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MATH2102 - REAL ANALYSIS

Problem Set 1

AUTUMN SEMESTER 2025-2026

Deadline: 15:00, Tuesday 21/10/2025

Your neat, clearly-legible solutions should be submitted electronically as a pdf file via the MATH2102 Moodle page by the deadline indicated there. A scan of a handwritten solution is acceptable. As this work is assessed, your submission must be entirely your own work (see the University’s policy on Academic Misconduct).

Submissions up to five working days late will be subject to a penalty of 5% of the maximum mark per working day.

The assessed questions for this Problem Set are Questions 2, 3, 4, 5.

Be aware that we may employ methods to detect AI use.

1. Let (an)n be a sequence of real numbers such that for all n,  Show that an → 0 as n → ∞.

Assessed questions start

2. (a) Let (rn)n be a sequence of natural numbers with rn → ∞ as n → ∞ and let a ∈ R. Show that there exists a sequence (pn)n∈N of integers with  → a as n → ∞. [15 marks]

(b) Let c ∈ R. Show that there exists a sequence (sn)n∈N of integers such that  → c as n → ∞.    [15 marks]

3. Consider for x ∈ R the sequence  Determine for each b ∈ R ∪ {∞} the set Sb = {x ∈ R ∶ fn(x) → b as n → ∞}.       [20 marks]

4. Let (xn)n∈N be a sequence with the following property:

5. (a) Let (an)n be a sequence and A ⊆ R its set of limit points. Show that if (an)n is bounded and f ∶ R → R is continuous, then the set of limit points of the sequence (f(an))n is f(A). [15 marks]

(b) Give an example of a sequence and a continuous function such that the set of limit points of (an)n is some set A ⊆ R, but the set of limit points of (f(an))n is not equal to f(A). [15 marks]

6. (a) Let r > 0. Using xy = exp(y log y) and the continuity of exp, or otherwise, show that  → 1 as n → ∞.

(b) Let k ∈ N and 0 < a1 ≤ a2 ≤ … ≤ ak for real numbers a1 , … , ak. Using the squeeze rule or otherwise, show that the sequence (bn)n given by

converges to ak as n → ∞.

7. (a) Let (an)n be a bounded sequence and let L be the set of limit points of (an)n. If (bk)k is a sequence with bk → b and bk ∈ L, show that b ∈ L.

(b) For the following sets Sj , j = 1, 2, 3, determine whether there exists a sequence that has the given set as limit points. Either give a sequence with the required properties or prove that no such sequence exists.

S1 = [0, 1), S2 = N, S3 = [0, 1]