Math 317 HW #2 Solutions
1. Exercise 1.3.3.
(a) Let A be bounded below, and dene B = cfw_b R : b is a lower bound for A. Show that
sup B = inf A.
Proof. First, note that B is bounded above by every element of A, so, by the Axiom of
Completeness, B has a s
ASSIGNMENT 1 SOLUTIONS
MAT 472 FALL 2011
Problem 1 (Exercise 1.2.2). True or False? If false, provide a counter-example.
(a) If A1 A2 A3 are innite sets, then An is innite.
n=1
(b) If A1 A2 A3 are nite non-empty sets, then An is nite and non-empty.
n=1
(c
2.2.1. Verify, using the denition of convergence of a sequence, that the
following sequences converge to the proposed limit.
1
= 0.
+1
Given a positive number " > 0 we have to nd a natural number N
1
so that for each n
N we have
< ". This inequality is
2+
Preface
My primary goal in writing Understanding Analysis was to create an elementary one-semester book that exposes students to the rich rewards inherent in taking a mathematically rigorous approach to the study of functions of a real variable. The aim o
Solutions to Week 2 Homework problems from Abbott
Problems (section 1.2) 1.3.1, 1.3.4, 1.3.6, 1.3.9, 1.4.2, 1.4.5, 1.4.11, 1.5.8
1.3.1 Note: This exercise is good practice for abstract algebra too!
(a) Show that, given any element z Z5 , there exists an e
ASSIGNMENT 2 SOLUTIONS
MAT 472 FALL 2011
Problem 1 (Exercise 1.3.4). Suppose A and B are non-empty subsets of R which are both bounded
above, and such that B A. Prove that sup B sup A.
Proof. Since a sup A for all a A and B A, we certainly have b sup A fo
1. For both of the following, assume that A, B R and that neither are empty. (10pts) (a) Show that A B = sup A sup B . Solution. If B is not bounded above, then sup B = , and it is trivial. So let b := sup B < and a := sup A. Then b is an upper bound for
MAT 472: INTERMEDIATE REAL ANALYSIS
FALL 2005
Instructor: S. Kaliszewski
Time: 10:4011:55 TTh / 3:154:30 TTh
Location: ARCH 321 / EDB 212
Line Number: 52462 / 49027
Credits: 3
Prerequisites: MAT 300 (Mathematical Structures) and MAT 342 (Linear Algebra),