Math 501
Introduction to Real Analysis
Instructor: Alex Roitershtein
Iowa State University
Department of Mathematics
Summer 2015
Exam #2
Sample
This is a take-home examination. The exam includes 6 questions. The total mark is 100
points. Please show all the work, not only the answers.
1. [17 points]
(a) Solve Exercise 40(c) in Chapter 2 of the textbook.
(b) Solve Exercise 40(d) in Chapter 2 of the textbook.
2. [17 points]
(a) Solve Exercise 78(a) in Chapter 2 of the textbook.
(b) Solve Exercise 78(c) in Chapter 2 of the textbook.
3. [17 points] Miscellaneous (and not necessarily related each to other) problems.
(a) Construct a compact set of real numbers such that its limit points form a countable
subset of R.
(b) Let (pn )n∈N be a Cauchy sequence of points in a metric space (X, d). Suppose that
some subsequence (pnk )k∈N of pn converges. Prove that then pn converges to the same
limit.
(c) Solve Exercise 54 in Chapter 2 of the textbook.
(d) Solve Exercise 71 in Chapter 2 of the textbook.
4. [16 points] Let A1 , A2 , . . . be subsets of a metric space. Let
Un = ∪nk=1 Ak
and
U = ∪∞
k=1 Ak .
(a) Prove that Un = ∪nk=1 Ak , n ∈ N.
(b) Prove that U ⊃ ∪∞
k=1 Ak .
(c) Show, by example, that the inclusion U ⊃ ∪∞
k=1 Ak might be proper.
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5. [17 points] Solve Exercise 55 in Chapter 2 of the textbook.
6. [16 points] Let (R, d) be a metric space where R = (0, +∞] and the metrics d is defined
as follows:
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d(x, y) = − x y
Let N∞ = {+∞, 1, 2, . . .} be the set including all positive integers together with +∞.
(a) Describe geometrically open balls Mr (x) in (R, d).
(b) Describe geometrically bounded sets in (R, d).
(c) Describe all interior, all cluster, and all isolated points of N∞ in (R, d) (a point p of a
subset S of a metric space is called isolated if it belongs to S and there exists r > 0
such that Mr (p) ∩ S contains only one point, namely p itself).
(d) Is N∞ open in (R, d)? Closed? Perfect (a set S in a metric space is called perfect if
any point of S is its cluster point)? Dense? Bounded?
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