MATH 3150 HOMEWORK 2
NOTE: All sequences and sets in this homework are assumed to be in R.
Problem 1 Let xn be a monotone increasing sequence such that xn+1 xn 1/n. Must
xn converge? HINT: Consider the sequence xn = 1 + 1/2 + + 1/n and you may use Prop
1.

MATH 3150 HOMEWORK 6
NOTE Unless otherwise specified in this assignment, Rn will denote Euclidean space with
its usual metric, norm and inner product.
Problem 1 Prove the following for subsets A and B of a metric space:
(a) int(int(A) = int(A).
(b) int(A)

MATH 3150 HOMEWORK 1
Problem 1. Prove the following proposition.
Proposition. In an ordered field, the following properties hold:
(i) Unique identities. If a + x = a for every a, then x = 0. If a x = a for
every a, then x = 1.
(ii) Unique inverses. If a +

MATH 3150 HOMEWORK 3
Problem 1 (p. 52, # 4)
(a) Let xn be a Cauchy sequence. Suppose that for every > 0 there is some n > 1/ such
that |xn | < . Prove that xn 0.
(b) Show that the hypothesis that xn be Cauchy is necessary, by coming up with an example
of

MATH 3150 HOMEWORK 5
NOTE Unless otherwise specified in this assignment, Rn will denote Euclidean space with
its usual metric, norm and inner product.
Problem 1 Let B Rn be any set. Define,
C = cfw_x Rn | d(x, y) < 1 for some y B.
Show that C is open.
Pro

MATH 3150 HOMEWORK 7
Problem 1 Let A (M, d) be a subset of a metric space. Show that if M is complete and
A is totally bounded, then cl(A) is compact.
Problem 2 Let (M, d) be a metric space and xn M a sequence such that xn x M .
Define,
A = cfw_x1 , x2 ,

MATH 3150 HOMEWORK 8
Problem 1
(a) Prove the following gluing lemma:
Let f : [a, b] Rm and g : [b, c] Rm be continous maps such that f (b) = g(b). Define
h : [a, c] Rm by h(t) = f (t) for t [a, b] and h(t) = g(t) for t [b, c]. Then h is continuous
on [a,