MIT18_100BF10_pset4

# MIT18_100BF10_pset4 - ∞ 4 Let a n n =1 be a sequence in R...

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18.100B : Fall 2010 : Section R2 Homework 4 Due Tuesday, October 5, 1pm Reading: Tue Sept.28 : connected sets, convergence, Rudin 2.45-47, 3.1-7 Thu Sept.30 : no reading assingment due to Quiz 1 (covering Rudin 1.1-38, 2.1-44) 1 . Consider the notes titled Compactness vs. Sequentially Compactness posted on the web page. Prove Lemma 3 stated on those notes. 2 . Assume ( X, d ) is a connected metric space. Prove that the only subsets that are both open and closed are X and . 3 . If in a metric space ( X, d ) we have B A X , then the set B is a connected subset of ( A, d ) (i.e. A with the relative topology) if and only if B is connected subset of ( X, d

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Unformatted text preview: ) . ) ∞ 4 . Let ( a n n =1 be a sequence in R with the property that no subsequence converges. Prove that | a n | → ∞ . Does the same property hold if the a n are in Q and we consider ( a n ) ∞ n =1 as a sequence in the metric space Q ? MIT OpenCourseWare http://ocw.mit.edu 18.100B Analysis I Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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## This document was uploaded on 10/26/2011 for the course MATH 18.100B at MIT.

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MIT18_100BF10_pset4 - ∞ 4 Let a n n =1 be a sequence in R...

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