Homework 05

Homework 05 - Q . Show there is one and only one way of...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 74 HOMEWORK 5 Due Friday March 8th at 3:10pm. Throughout, let X and Y be metric spaces. (1) Let { x n } n =0 be a Cauchy sequence, and suppose that a subsequence { x n i } i =0 converges. Show that the whole sequence { x n } must converge. (2) Show that x A if and only if inf a A d ( x,a ) = 0. (3) Show that if x n x then d ( x n ,x m ) 0 as n,m → ∞ . (4) Let d 1 and d 2 be two metrics on the same set X . Further suppose that d 1 ( x,y ) λd 2 ( x,y ) for some positive real number λ . Show that if a sequence { x n } converges with respect to d 2 , then it converges with respect to d 1 . (5) Let f : Q Q be a continuous function with respect to the usual metric on
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Q . Show there is one and only one way of extending it to the reals as a continuous function ˜ f : R → R . (6) Show that a Cauchy sequence is bounded (as a set). Conclude that unbounded sequences cannot converge. (7) Find an example of a bounded set B and a continuous function f : R → R such that f ( B ) is not bounded. (8) Show that R is not compact. (9) Show that a bounded set in R is totally bounded. 1...
View Full Document

This note was uploaded on 05/04/2011 for the course MATH 73 taught by Professor Danberwick during the Spring '09 term at Berkeley.

Ask a homework question - tutors are online