Real Analysis Solutions Problem Set 5
Drew D. Ash
May 13, 2013
pg. 83 Exercise 3.2.2: Let
B=
(1)n n
: n = 1, 2, 3, . . .
n+1
(a) Find the limit points of B .
(b) Is B a closed set?
(c) Is B an open set?
(d) Does B contain any isolated points?
(e) Find B .
Real Analysis Problem Set 4 Solutions
Drew D. Ash
April 30, 2013
pg. 49 Exercise 2.3.7 part a: Let (an ) be a bounded (not necessarily convergent) sequence, and assume
lim bn = 0. Show that lim(an bn ) = 0. Why are we not allowed to use the Algebraic Limi
Real Analysis Problem Set 7 Solutions
Drew D. Ash
May 29, 2013
pg. 113 Exercise 4.3.2; Part (b): Give a proof of Theorem 4.3.9 using the sequential characterization
of continuity.
Proof. Recall,
Theorem 4.3.9 : Given f : A R and g : B R, assume that the r
MATH 3161 Practice Midterm Exam
Instructions: You may not use any instructional aids (book, notes, calculator,
etc.) on this exam. For problems which require proof, you should prove everything either from basic building blocks of the real numbers or clear
Real Analysis Problem Set 8 Solutions
Drew D. Ash
May 29, 2013
pg. 161 Exercise 6.2.3: Consider the sequence of functions
hn (x) =
x
1 + xn
over the domain [0, ).
(a) Find the pointwise limit of (hn ) on [0, ).
(b) Explain how we know that the convergence
MATH 3161 Practice Final Exam Solutions
1. For each of the following, either provide an example or use a theorem or fact from class to explain
why such an example cannot exist. You DO NOT need to show work for your examples!
(a) A countable closed set
Sol
MATH 3161 Practice Final Exam
1. For each of the following, either provide an example or use a theorem or fact from class to explain
why such an example cannot exist. You DO NOT need to show work for your examples!
(a) A countable closed set
(b) A countab
MATH 3161 Practice Midterm Exam Solutions
Instructions: You may not use any instructional aids (book, notes, calculator,
etc.) on this exam. For problems which require proof, you should prove everything either from basic building blocks of the real number
Real Analysis Problem Set 6 Solutions
Drew D. Ash
May 22, 2013
pg. 87 Exercise 1: Show that if K is compact, then sup K and inf K both exists and are elements of K .
Proof. Let K be a compact subset of R. Since K is compact, K is both closed and bounded.
Solutions for Problem Set 1: Real Analysis
Drew D. Ash
April 9, 2013
pg. 11 Exercise 1.2.2 part a: Prove or disprove the following: If A1 A2 A3 . . . are all sets
containing an innite number of elements, then the intersection,
n=1
An
is innite as well.
Pr
Real Analysis Problem Set 2 Solutions.
Drew D. Ash
April 16, 2013
pg. 11 Exercise 1.2.7 part b: Show that for an arbitrary function g : R R, it is always true that
g 1 (A B ) = g 1 (A) g 1 (B ) and g 1 (A B ) = g 1 (A) g 1 (B )
for all sets A, B R.
Proof.
Real Analysis Problem Set 3 Solutions
Drew D. Ash
April 22, 2013
pg. 43 Exercise 2.2.1 Part b: Verify, using the denition of convergence of a sequence, that the following
sequence converges to the proposed limit.
3
3n + 1
=
lim
n 2n + 5
2
Proof. We begin