MATH40002 Set2 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: 85e7a47e8596557d9c79e081f07121edbd81ab4233665b48cbcd17583b5679c8Source 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.
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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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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.
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First key step
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(a) First find a binary-choice subset whose sequences converge automatically. (d) First make |a_n| ≤ 1/k, then use the first crossing.
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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 2
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
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Common errors
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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.
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Final self-check
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Check that every diagonal-sequence term is rational and the sequence tends to 1; ensure the crossing indices n_k are strictly increasing.
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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
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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.
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First key step
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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.
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Two definitions: 6 marks; Cauchy implies bounded: 4; product limit: 5; unique cluster-point method: 5.
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: 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.
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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.
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For weighted sums, introduce the series tails and use summation by parts.
English passage 11 (51% down the source page)
First key step
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Respectively: use the contrapositive; exclude a finite preimage; calculate b_(n+1) − b_n; define r_j as a tail sum.
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 (12% down the source page)
Contrapositive and comparison: 5 marks; finite preimage: 5; monotone boundedness: 5; tail-sum identity: 5.
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Common errors
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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.
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Final self-check
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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.
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For a one-point integrability change use a spike. For a nonnegative continuous function with zero integral, use a local positive lower bound.
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First key step
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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 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 (17% down the source page)
Four judgements: 4 marks; one-sided limit: 5; critical point: 4; one-point integrability change: 4; nonnegative integral: 3.
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Common errors
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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.