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 th
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
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 =
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 b
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
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
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