MATH40002 6 Sets REVISED
English review edition prepared on 4 October 2026 from a preserved source copy. It is a later presentation, not the historical study interface. Source checks and difficulty judgements describe the original material author's own process; they do not indicate Imperial College London endorsement.
Source SHA-256: 8e62949ae40d5e0b970da5fdaba3daa6f3bc61c1aef4d8ee2e6af84c8a56556aSource date: 2026-08-06
Source page 1
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (19% down the source page)
The main solutions prioritise methods actually used in the official answers accessible during preparation.
English passage 2 (21% down the source page)
No lecture notes, problem sheets, tutorials or marked scripts from Tony were accessible. This booklet does not invent their contents, and advanced alternative methods are not the sole standard for solutions.
English passage 3 (37% down the source page)
Recognition signal
English passage 4 (40% down the source page)
The same positive sequence appears in sequence limits, ordinary series, alternating series and power series. First look for explicit differences, monotonicity and partial sums.
English passage 5 (45% down the source page)
First key step
English passage 6 (48% down the source page)
First write a_n = log(1 + 1/n). Immediately check whether the ordinary series telescopes; on the unit circle, check whether the geometric partial sums are bounded.
Source page 2
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (12% down the source page)
Monotonicity and limit: 3 marks; use MVT to locate ξ_n and complete the ε–N proof: 3; telescoping partial sums: 3; alternating test and remainder: 5; radius and boundary analysis:
English passage 2 (14% down the source page)
Dirichlet 6 marks .
English passage 3 (17% down the source page)
Common errors
English passage 4 (20% down the source page)
Common mistakes: merely write a_n ∼ 1/n without the required ε–N proof; incorrectly claim absolute convergence on |z| = 1; fail to show that the geometric partial sums are bounded when z ≠ 1.
English passage 5 (25% down the source page)
Final self-check
English passage 6 (27% down the source page)
check PN
English passage 7 (28% down the source page)
n=1 an = log(N + 1); check that the boundary absolute value remains an.
English passage 8 (31% down the source page)
Tony study template card
English passage 9 (34% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 10 (36% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 11 (52% down the source page)
Recognition signal
English passage 12 (55% down the source page)
Values specified on a dense set suggest choosing a sequence from that set. Finite limits at open endpoints suggest extending to the endpoints and using Heine–Cantor.
English passage 13 (57% down the source page)
Rapid oscillation suggests using two sequences to disprove uniform continuity.
English passage 14 (60% down the source page)
First key step
English passage 15 (63% down the source page)
Respectively: take a sequence from the dense set; define the endpoint values; construct two sequences whose distance tends to zero but whose function values differ by a fixed amount.
Source page 3
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (19% down the source page)
Density and continuity: 5 marks; endpoint extension and proof at both ends: 4; Heine–Cantor: 4; continuous boundedness: 2; two-sequence counterexample: 5.
English passage 2 (22% down the source page)
Common errors
English passage 3 (24% down the source page)
Common mistakes: state density of the rationals/irrationals without choosing convergent sequences; assume continuity automatically gives uniform continuity; choose two sequences with equal function values, or whose distance does not tend to zero.
English passage 4 (30% down the source page)
Final self-check
English passage 5 (32% down the source page)
Check that both sequences lie in the domain, approach the same endpoint, and have function values differing by exactly 2.
English passage 6 (36% down the source page)
Tony study template card
English passage 7 (38% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 8 (40% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 9 (56% down the source page)
Recognition signal
English passage 10 (59% down the source page)
For true/false questions, first check closure properties, convex/concave inequality directions, the chain rule, and whether twice differentiability requires a first derivative throughout a neighbourhood. State every theorem hypothesis.
English passage 11 (65% down the source page)
First key step
English passage 12 (67% down the source page)
First try simple affine, densely piecewise or oscillatory counterexamples. For an inverse-function question, start with h ◦ f = id.
Source page 4
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (36% down the source page)
True/false conclusions with reasons: 12 marks; three precise definitions or theorem statements: 8.
English passage 2 (39% down the source page)
Common errors
English passage 3 (41% down the source page)
Common mistakes: give True/False without proof; incorrectly extend preservation of convexity under maxima to concavity; infer twice differentiability merely from existence of u′(0); use the wrong order or coefficient in the Taylor remainder.
English passage 4 (47% down the source page)
Final self-check
English passage 5 (49% down the source page)
Check each theorem's domain, orders of continuity and differentiability, quantifier order, and integral-remainder coefficient.
English passage 6 (53% down the source page)
Tony study template card
English passage 7 (56% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 8 (58% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 9 (74% down the source page)
Recognition signal
English passage 10 (76% down the source page)
Finitely many bad points suggest covering them by intervals of small total length. A fixed second-derivative sign and a bounded function suggest a monotone derivative.
English passage 11 (78% down the source page)
Rule out a nonzero slope by contradiction, then use an integral squeeze.
English passage 12 (82% down the source page)
First key step
English passage 13 (84% down the source page)
For integrability, first write |f| ≤ M and split into good and bad regions. For convex/concave asymptotics, first use f′′ to determine the monotonicity of f′.
Source page 5
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (47% down the source page)
Bad-region length and uniform continuity on good regions: 4 marks; monotonicity: 3; a supremum-definition proof of the limit: 3; derivative limit: 4; integral squeeze: 4;
English passage 2 (49% down the source page)
final squeeze 2 marks .
English passage 3 (52% down the source page)
Common errors
English passage 4 (54% down the source page)
Common mistakes: omit interval length in the bad-region oscillation estimate; incorrectly infer that f′ increases from f′′ ≤ 0; use a supremum without its approximation property; finally infer the product limit from the derivative limit alone.
English passage 5 (56% down the source page)
f ′(x) →0 directly infer xf ′(x) →0
English passage 6 (60% down the source page)
Final self-check
English passage 7 (63% down the source page)
Check every inequality direction against concavity. The final squeeze uses the function difference tending to zero, not the product-limit theorem.
English passage 8 (66% down the source page)
Tony study template card
English passage 9 (69% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 10 (71% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
Source page 6
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (19% down the source page)
The main solutions prioritise methods actually used in the official answers accessible during preparation.
English passage 2 (21% down the source page)
No lecture notes, problem sheets, tutorials or marked scripts from Tony were accessible. This booklet does not invent their contents, and advanced alternative methods are not the sole standard for solutions.
English passage 3 (37% down the source page)
Recognition signal
English passage 4 (40% down the source page)
The key 2025 methods are Cantor diagonalisation, nonnegative subseries, extracting a subsequence from infinitely many visits to a bounded interval, and crossing an interval with small steps.
English passage 5 (45% down the source page)
First key step
English passage 6 (48% down the source page)
(a) First find a binary-choice subset whose sequences converge automatically. (d) First make |a_n| ≤ 1/k, then use the first crossing.
English passage 7 (90% down the source page)
Diagonal subset: 6 marks; upper bound for partial sums selected by an injection: 4; bounded subsequence and Bolzano–Weierstrass: 3; first-crossing construction: 7.
Source page 7
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Common errors
English passage 2 (12% down the source page)
Common mistakes: a diagonal sequence that need not converge; calling an injection a permutation; describing repeated crossings without the first-crossing construction and error estimate.
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
Check that every diagonal-sequence term is rational and the sequence tends to 1; ensure the crossing indices n_k are strictly increasing.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (24% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 7 (26% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 8 (43% down the source page)
Recognition signal
English passage 9 (45% down the source page)
Start definition questions with quantifiers. For Cauchy implies bounded, take ε = 1. For all convergent subsequences having the same limit, use contradiction and Bolzano–Weierstrass.
English passage 10 (49% down the source page)
First key step
English passage 11 (52% down the source page)
Write the full definitions. For a product limit, first establish eventual boundedness of one factor. For a unique subsequential limit, construct a subsequence that stays away from the target.
English passage 12 (85% down the source page)
Two definitions: 6 marks; Cauchy implies bounded: 4; product limit: 5; unique cluster-point method: 5.
Source page 8
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Common errors
English passage 2 (12% down the source page)
Common mistakes: stating the Cauchy definition only for adjacent terms; dividing by a possibly zero quantity in a product-limit proof; failing to explain how bad points form a subsequence in the contradiction argument.
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
Check that the statement holds for all indices after N, and that the product decomposition is correct.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (25% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 7 (26% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 8 (43% down the source page)
Recognition signal
English passage 9 (45% down the source page)
Use the contrapositive for conditional convergence. For permuted sequence limits, the preimage of finitely many bad indices remains finite. For harmonic windows, first try monotonicity and boundedness.
English passage 10 (47% down the source page)
For weighted sums, introduce the series tails and use summation by parts.
English passage 11 (51% down the source page)
First key step
English passage 12 (53% down the source page)
Respectively: use the contrapositive; exclude a finite preimage; calculate b_(n+1) − b_n; define r_j as a tail sum.
Source page 9
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (12% down the source page)
Contrapositive and comparison: 5 marks; finite preimage: 5; monotone boundedness: 5; tail-sum identity: 5.
English passage 2 (15% down the source page)
Common errors
English passage 3 (17% down the source page)
Incorrectly write (∑ a_n)^2 = ∑ a_n^2.
English passage 4 (18% down the source page)
Other mistakes: assume every bijection is monotone; seek a harmonic-window limit before proving convergence; use incorrect tail-sum indices.
English passage 5 (23% down the source page)
Final self-check
English passage 6 (26% down the source page)
Check r_j − r_(j+1) = a_j; verify the window-difference formula for small n.
English passage 7 (29% down the source page)
Tony study template card
English passage 8 (32% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 9 (34% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 10 (52% down the source page)
Recognition signal
English passage 11 (55% down the source page)
For true/false questions check endpoint hypotheses. Use supremum approximation for one-sided monotone limits; use the Darboux property of derivatives to find a critical point when endpoint limits agree.
English passage 12 (57% down the source page)
For a one-point integrability change use a spike. For a nonnegative continuous function with zero integral, use a local positive lower bound.
English passage 13 (60% down the source page)
First key step
English passage 14 (63% down the source page)
Start with the key tool in each part: supremum approximation, the intermediate-value property of derivatives, g = f + h, or continuity giving a positive interval.
Source page 10
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (17% down the source page)
Four judgements: 4 marks; one-sided limit: 5; critical point: 4; one-point integrability change: 4; nonnegative integral: 3.
English passage 2 (20% down the source page)
Common errors
English passage 3 (22% down the source page)
Common mistakes: add unstated endpoint continuity in (iii) or (iv); assume the one-sided limit exists; claim upper and lower sums are unchanged by a one-point modification.
English passage 4 (26% down the source page)
Final self-check
English passage 5 (28% down the source page)
Check that every counterexample satisfies the stated domain and differentiability assumptions.
English passage 6 (32% down the source page)
Tony study template card
English passage 7 (35% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 8 (37% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
Source page 11
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (18% down the source page)
The main solutions prioritise methods actually used in the official answers accessible during preparation.
English passage 2 (20% down the source page)
No lecture notes, problem sheets, tutorials or marked scripts from Tony were accessible. This booklet does not invent their contents, and advanced alternative methods are not the sole standard for solutions.
English passage 3 (37% down the source page)
Recognition signal
English passage 4 (39% down the source page)
For integer-valued monotone functions, ask whether they are eventually constant. Try diagonalisation for uncountability. For a supremum, prove both upper-boundedness and leastness; for reciprocal limits keep the denominator away from zero.
English passage 5 (45% down the source page)
First key step
English passage 6 (47% down the source page)
Respectively: stratify by eventual constancy; assume countability and diagonalise; first prove the upper bound; first ensure |a_n| > |a|/2.
English passage 7 (80% down the source page)
Countably many nonincreasing functions: 5 marks; uncountably many nondecreasing functions: 5; supremum of products: 5; reciprocal limit: 5.
English passage 8 (83% down the source page)
Common errors
English passage 9 (86% down the source page)
Common mistakes: call all functions from N to a finite set countable; fail to make the diagonal construction monotone; prove an upper bound without leastness; divide by an uncontrolled a_n in the reciprocal proof.
Source page 12
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Final self-check
English passage 2 (12% down the source page)
Check that the diagonal function belongs to T and that the supremum proof uses positivity.
English passage 3 (16% down the source page)
Tony study template card
English passage 4 (18% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 5 (20% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 6 (38% down the source page)
Recognition signal
English passage 7 (41% down the source page)
When a downward drop is bounded by a summable error, add or subtract tail terms to make a monotone sequence. For fractional recurrences calculate adjacent differences; for averages first split into a finite head and a small tail.
English passage 8 (46% down the source page)
First key step
English passage 9 (49% down the source page)
Define a corrected sequence c_n; directly subtract recurrence relations. For Cesàro averages, first fix the tail threshold.
English passage 10 (88% down the source page)
Corrected monotone sequence: 5 marks; contracting differences, Cauchy property and limit: 7; head/tail split: 6; counterexample: 2.
Source page 13
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Common errors
English passage 2 (12% down the source page)
Common mistakes: claim the recurrence sequence is monotone; solve for a fixed point without proving convergence; treat every term as small in the Cesàro proof.
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
The recurrence has positive and negative fixed-point roots. Use positivity to select the correct root.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (25% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 7 (26% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 8 (43% down the source page)
Recognition signal
English passage 9 (45% down the source page)
Arbitrary sequences in a function-limit or continuity question suggest the sequential criterion. For rational/irrational pieces use sequences from both dense sets; for fixed points use IVT.
English passage 10 (51% down the source page)
First key step
English passage 11 (54% down the source page)
Start with 0 < |x − a| in the definition. For the converse, negate the definition and construct |x_n − a| < 1/n.
English passage 12 (85% down the source page)
Definition: 2 marks; sequential criterion: 6; piecewise limit: 4; Dirichlet: 3; squeeze: 2; fixed points: 3.
Source page 14
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Common errors
English passage 2 (12% down the source page)
Common mistakes: fail to exclude x = a in a limit definition; prove only one direction of the sequential criterion; omit endpoint-sign checks for a fixed point.
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
At a = 0 both branches tend to 0; for a ≠ 0 the two limits really are distinct.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (24% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 7 (26% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 8 (43% down the source page)
Recognition signal
English passage 9 (45% down the source page)
One-point modification suggests a spike; monotone integrability suggests telescoping upper/lower sums on uniform partitions; finite limits at infinity suggest bounded tails and a compact middle interval. Equal limits at both ends suggest a critical point.
English passage 10 (51% down the source page)
First key step
English passage 11 (53% down the source page)
Write the one-point difference as h; calculate U − L directly for a monotone function; split the real line into three pieces to prove boundedness.
English passage 12 (86% down the source page)
One-point modification: 4 marks; upper/lower sums and telescoping: 8; boundedness on three pieces: 4; critical point: 4.
Source page 15
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Common errors
English passage 2 (12% down the source page)
Common mistakes: fail to control the interval around the changed point; reverse upper and lower endpoints for a monotone function; forget the compact middle after bounding the tails.
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
U − L must equal the endpoint difference divided by n.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (24% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 7 (26% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
Source page 16
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (19% down the source page)
The main solutions prioritise methods actually used in the official answers accessible during preparation.
English passage 2 (20% down the source page)
No lecture notes, problem sheets, tutorials or marked scripts from Tony were accessible. This booklet does not invent their contents, and advanced alternative methods are not the sole standard for solutions.
English passage 3 (39% down the source page)
Recognition signal
English passage 4 (42% down the source page)
For cardinality classify infinitely many free coordinates, eventually fixed coordinates, and fixed tails. Read quantifiers layer by layer; prove a supremum in two steps; choose an unbounded subsequence whose absolute values successively exceed k.
English passage 5 (47% down the source page)
First key step
English passage 6 (50% down the source page)
Write T = ∪U ; for sup prove an upper bound and then ε-approximation; recursively choose indices for the unbounded subsequence.
English passage 7 (75% down the source page)
Cardinality classification: 5 marks; interpreting quantifiers: 4; supremum proof: 4; unbounded subsequence: 7.
English passage 8 (78% down the source page)
Common errors
English passage 9 (81% down the source page)
Common mistakes: assume all infinite sequences over a finite alphabet form a countable set; misread whether a depends on N; select only values above k and ignore negative unboundedness.
English passage 10 (86% down the source page)
Final self-check
English passage 11 (89% down the source page)
For unboundedness use |a_n|. U_N has N − 1 free coordinates.
Source page 17
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Tony study template card
English passage 2 (12% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 3 (14% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 4 (30% down the source page)
Recognition signal
English passage 5 (33% down the source page)
Obtain a telescoping lower bound by rationalising a square-root difference. For exponential terms with a_n → a < 1, use eventual geometric comparison. Split a power series defined by parity into its even and odd subseries.
English passage 6 (38% down the source page)
First key step
English passage 7 (41% down the source page)
First rationalise; choose A ∈ (a, 1); separate the even and odd terms.
English passage 8 (76% down the source page)
Square-root inequality: 2 marks; divergence from first principles: 6; eventual geometric comparison: 6; ordinary series and radius: 6.
English passage 9 (79% down the source page)
Common errors
English passage 10 (81% down the source page)
Common mistakes: reverse an inequality; cite the p-series test despite the first-principles requirement; misread the exponent on a_n.
English passage 11 (81% down the source page)
Do not mistake a_n^n for n a_n, or apply the alternating test solely because an expression alternates.
English passage 12 (85% down the source page)
Final self-check
English passage 13 (88% down the source page)
Divergence at z = 1 suffices for R ≤ 1; inside the radius, absolute convergence must be proved.
Source page 18
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Tony study template card
English passage 2 (12% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 3 (14% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 4 (32% down the source page)
Recognition signal
English passage 5 (35% down the source page)
For oscillation multiplied by a power at 0, use squeezing and the difference quotient; for a sign change use IVT to obtain a root; the second derivative sign gives convex/concave midpoint inequalities; for multiple zeros repeatedly apply
English passage 6 (40% down the source page)
First key step
English passage 7 (43% down the source page)
First bound | sin | ≤ 1; construct a continuous difference function; calculate the sign of h; decrease the zero count level by level.
English passage 8 (75% down the source page)
Continuity/differentiability: 5 marks; IVT and zeros: 5; Taylor + concavity: 6; repeated Rolle: 4.
English passage 9 (78% down the source page)
Common errors
English passage 10 (81% down the source page)
Common mistakes: apply the chain rule directly at x = 0; allow endpoint zeros despite an open-interval requirement; reverse the convex/concave inequality; undercount zeros when repeatedly using Rolle.
English passage 11 (86% down the source page)
Final self-check
English passage 12 (89% down the source page)
h should be concave, so its midpoint value exceeds the average endpoint value.
Source page 19
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Tony study template card
English passage 2 (12% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 3 (14% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 4 (30% down the source page)
Recognition signal
English passage 5 (33% down the source page)
For Darboux sums on a special partition use monotonicity to choose endpoints; compare improper integrals with 1/xr; for power-series orthogonality substitute the truncated polynomial
English passage 6 (35% down the source page)
and then use uniform convergence to obtain the integral of the square.
English passage 7 (38% down the source page)
First key step
English passage 8 (41% down the source page)
write xi = ei/n; use ey ≥1 + y and MVT to squeeze; let PN is Taylor truncation .
English passage 9 (74% down the source page)
Upper and lower sums: 8 marks; improper integral: 6; power-series orthogonality: 6.
English passage 10 (77% down the source page)
Common errors
English passage 11 (80% down the source page)
Common mistakes: forget that the Darboux interval lengths are not a constant 1/n; state asymptotic inequalities for r > 1 without proof; interchange integral and limit without explaining uniform convergence.
English passage 12 (85% down the source page)
Final self-check
English passage 13 (88% down the source page)
U −L should simplify to (e −1)/n;
Source page 20
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Tony study template card
English passage 2 (12% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 3 (14% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
Source page 21
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (19% down the source page)
The main solutions prioritise methods actually used in the official answers accessible during preparation.
English passage 2 (21% down the source page)
No lecture notes, problem sheets, tutorials or marked scripts from Tony were accessible. This booklet does not invent their contents, and advanced alternative methods are not the sole standard for solutions.
English passage 3 (38% down the source page)
Recognition signal
English passage 4 (40% down the source page)
For rearrangements return to partial sums; for a tail r first write a = r − r ; a square-root denominator suggests rationalisation and telescoping; for a /r
use blocks in the Cauchy criterion.
English passage 5 (46% down the source page)
First key step
English passage 6 (48% down the source page)
Begin the definition with “all sufficiently late partial sums”; for a tail, first write the difference identity.
English passage 7 (88% down the source page)
Definition: 3 marks; rearrangement: 7; square-root tail: 4; Cauchy divergence: 6.
Source page 22
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Common errors
English passage 2 (12% down the source page)
Treating an infinite sum as a number whose terms may be rearranged freely; reversing the inequality between negative partial sums and their limit; trying termwise comparison with the harmonic series in (d).
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
r decreases; any finite sum of negative terms is at least the sum of all negative terms.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (25% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 7 (26% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 8 (43% down the source page)
Recognition signal
English passage 9 (45% down the source page)
Use neighbourhoods and sequences for open/closed sets. For openness of the image of a strictly monotone continuous function, use local endpoints and IVT. Subdivide and add estimates to control long intervals using uniform continuity. Disprove uniform continuity using two sequences.
English passage 10 (51% down the source page)
First key step
English passage 11 (53% down the source page)
For openness of the image, fix y = f (x); for a growth bound, first take ε = 1; for oscillation, choose a phase difference of π.
English passage 12 (85% down the source page)
Set classification: 5 marks; open mapping: 5; linear growth bound: 5; two-sequence method: 5.
Source page 23
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Common errors
English passage 2 (12% down the source page)
Assuming every continuous function is an open map; incorrectly making the number of subintervals independent of x; choosing two sequences with fixed phase difference but whose distance does not tend to zero.
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
Use both strict monotonicity and IVT; simplify x − y using a difference of squares.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (25% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 7 (26% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 8 (43% down the source page)
Recognition signal
English passage 9 (45% down the source page)
Use a standard collection of examples; repeatedly apply Rolle to equal values; first construct a difference function for MVT; read Taylor coefficients from the power series, and use the second-derivative sign for convexity.
English passage 10 (51% down the source page)
First key step
English passage 11 (53% down the source page)
Start with the simplest example; track the decreasing number of zeros under Rolle; rewrite two solutions of an equation as two zeros of a difference function.
English passage 12 (81% down the source page)
Examples: 6 marks; repeated Rolle: 4; three MVT applications: 6; power series + convexity: 4.
English passage 13 (83% down the source page)
Common errors
English passage 14 (86% down the source page)
Common mistakes: use a constant function as an example with no local extrema; apply Rolle once too few; reverse the arctan inequality; reverse the midpoint-convexity inequality.
Source page 24
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Final self-check
English passage 2 (12% down the source page)
F should be convex on (0, 1), so its midpoint value does not exceed the average of the endpoint values.
English passage 3 (16% down the source page)
Tony study template card
English passage 4 (18% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 5 (20% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 6 (38% down the source page)
Recognition signal
English passage 7 (41% down the source page)
For an ODE with zero endpoints, use a positive maximum/negative minimum; if the integral vanishes for all test functions, use a local tent function; for nested integral identities,
English passage 8 (43% down the source page)
compare derivatives and initial values; isolate a single-point spike with a narrow partition interval.
English passage 9 (46% down the source page)
First key step
English passage 10 (49% down the source page)
Assume a positive value exists and take a global maximum; explicitly choose a nonnegative test function that is locally positive; name both sides of the identity.
English passage 11 (74% down the source page)
Maximum principle: 5 marks; test functions: 6; integral identity: 5; integrability of a spike: 4.
English passage 12 (77% down the source page)
Common errors
English passage 13 (80% down the source page)
Forgetting the zero endpoints when considering where the maximum occurs; choosing a test function that fails the endpoint conditions; exchanging integration order without stating the conditions;
English passage 14 (82% down the source page)
Incorrectly calculate the spike upper sum as 1.
English passage 15 (85% down the source page)
Final self-check
English passage 16 (88% down the source page)
Repeat the argument for −f to obtain equality to 0; both sides of the identity vanish at x = 0.
Source page 25
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Tony study template card
English passage 2 (12% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 3 (14% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
Source page 26
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (19% down the source page)
The main solutions prioritise methods actually used in the official answers accessible during preparation.
English passage 2 (21% down the source page)
No lecture notes, problem sheets, tutorials or marked scripts from Tony were accessible. This booklet does not invent their contents, and advanced alternative methods are not the sole standard for solutions.
English passage 3 (39% down the source page)
Recognition signal
English passage 4 (42% down the source page)
Use an injection to prove a function space uncountable. For the supremum of quotients, first bound and then approximate. Split cases for the limit of a maximum. For summable upward errors, add the tail to construct a monotone sequence.
English passage 5 (47% down the source page)
First key step
English passage 6 (50% down the source page)
Construct fg; choose sequences approaching sup/inf; assume a ≥ b without loss of generality; define tn as the tail of a p-series.
English passage 7 (85% down the source page)
Uncountability: 4 marks; supremum of quotients: 5; maximum limit: 5; error-corrected sequence: 6.
Source page 27
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Common errors
English passage 2 (12% down the source page)
Common mistakes: interpret periodicity as simultaneous translation of both variables; fail to check whether inf B may be zero; use continuity of a composite when a first-principles proof is required; choose the wrong sign for the corrected sequence.
English passage 3 (17% down the source page)
Final self-check
English passage 4 (20% down the source page)
c_n must be nonnegative and monotonically decreasing; t_n − t_(n+1) = 1/n².
English passage 5 (24% down the source page)
Tony study template card
English passage 6 (26% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 7 (28% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 8 (45% down the source page)
Recognition signal
English passage 9 (47% down the source page)
For the radius of the coefficientwise product, use two interior radii and geometric comparison; terms of a convergent series are bounded; a positive-term average includes at least the first-term harmonic lower bound;
English passage 10 (49% down the source page)
squeeze the geometric mean of finitely many consecutive factors.
English passage 11 (53% down the source page)
First key step
English passage 12 (55% down the source page)
Choose x < R1, y < R2 and r < xy; Give |an| find a uniform bound; write bn ≥a1/n.
English passage 13 (87% down the source page)
Hadamard-product radius: 6 marks; weighted absolute comparison: 3; divergence of the series of averages: 4; geometric means and the alternating test: 7.
Source page 28
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Common errors
English passage 2 (12% down the source page)
Proving only that the general term tends to 0 rather than convergence of the power series; treating af(n) as though it were a subsequence; omitting the monotonicity condition of the alternating test.
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
The comparison ratio r/(xy) must be strictly less than 1.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (25% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 7 (27% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 8 (45% down the source page)
Recognition signal
English passage 9 (47% down the source page)
Use sup for a monotone one-sided limit; a difference function for a fixed point; the chain rule for the derivative of an even function; if a function has a finite limit and its derivative has a limit,
English passage 10 (49% down the source page)
argue by contradiction; use nearby large points to disprove uniform continuity of a polynomial.
English passage 11 (53% down the source page)
First key step
English passage 12 (56% down the source page)
Respectively write L = sup, h = f − x, f ′(x) = −f ′(−x), assume the derivative limit is nonzero, and choose n and n + 1/n.
English passage 13 (80% down the source page)
One-sided limit: 5 marks; fixed point: 3; even function: 3; derivative limit: 4; failure of uniform continuity: 5.
English passage 14 (83% down the source page)
Common errors
English passage 15 (85% down the source page)
Common mistakes: assume existence of the one-sided limit; use Rolle for an even function to obtain a zero derivative somewhere rather than at 0; assume f′ tends to zero automatically by interchanging limits.
Source page 29
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Final self-check
English passage 2 (12% down the source page)
For failure of uniform continuity, both points should tend to infinity while their distance tends to 0.
English passage 3 (16% down the source page)
Tony study template card
English passage 4 (18% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 5 (20% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 6 (36% down the source page)
Recognition signal
English passage 7 (39% down the source page)
Darboux sums with finitely many bad points, nonnegative integrals, uniform Taylor remainders, subtracting a chord and using Rolle, and asymptotics of integral averages are frequent methods from five different years.
English passage 8 (44% down the source page)
First key step
English passage 9 (47% down the source page)
For integrability split into good and bad regions; for Taylor state a remainder bound; for a chord subtract the linear function; for integral asymptotics rewrite the limit as a two-sided inequality.
English passage 10 (88% down the source page)
Integrability with finite discontinuities: 5 marks; nonnegative integral: 3; uniform Taylor convergence: 4; chord and Rolle: 4; integral asymptotics: 4.
Source page 30
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Common errors
English passage 2 (12% down the source page)
An off-by-one subscript in the Taylor remainder factorial; forgetting that endpoints are also chord intersections; ignoring the constant contribution from [0, N] in integral asymptotics.
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
The uniform Taylor error does not depend on x; after division by x2 the constant term tends to 0.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (25% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 7 (27% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.