18.100B.Homework3

18.100B.Homework3 - 18.100B/C: Fall 2008 Homework 3...

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Unformatted text preview: 18.100B/C: Fall 2008 Homework 3 Available Wednesday, September 24 Due Wednesday, October 1 Turn in the homework by 11am on Wednesday, October 1, in 2-108. For 18.100B it should be put in the bin corresponding to the lecture you regularly attend (regardless of which one you may be officially enrolled in). If you are enrolled in 18.100C, put your homework in the C bin in any case. (For 18.100C also recall that you need to write at least one of your problem solutions in L A T E X.) 1. (Principle of continuous induction) Let ( X, d ) be a connected metric space. Suppose E ⊂ X is a subset that has the follow- ing three properties: (a) There exists a point x ∈ E . (b) For every x ∈ E we can find r > such that the ball B r ( x ) ⊂ X is entirely con- tained in E . (c) For every sequence ( x n ) n ∈ N in E that converges in X , the limit point lim n →∞ x n lies in E . Show that E = X . (Hint: Homework 2, Problem 8.) 2. (Definitions of convergence) Let ( x n ) n ∈ N be a sequence in a metric space...
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This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.

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18.100B.Homework3 - 18.100B/C: Fall 2008 Homework 3...

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