MATH40002 Set3 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: 97766dd08536257cab49438a9b366bed05925afb55e4c35788c6f407ed5de68aSource 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 (18% down the source page)
The main solutions prioritise methods actually used in the official answers accessible during preparation.
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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.
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Recognition signal
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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.
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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
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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 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 (9% down the source page)
Final self-check
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Check that the diagonal function belongs to T and that the supremum proof uses positivity.
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Tony study template card
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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
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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.
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First key step
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Define a corrected sequence c_n; directly subtract recurrence relations. For Cesàro averages, first fix the tail threshold.
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Corrected monotone sequence: 5 marks; contracting differences, Cauchy property and limit: 7; head/tail split: 6; counterexample: 2.
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 (9% down the source page)
Common errors
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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
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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.
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First key step
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Start with 0 < |x − a| in the definition. For the converse, negate the definition and construct |x_n − a| < 1/n.
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Definition: 2 marks; sequential criterion: 6; piecewise limit: 4; Dirichlet: 3; squeeze: 2; fixed points: 3.
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 (9% down the source page)
Common errors
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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
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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 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 (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.