real-analysis-hw1-with-solution-2007-9-11

real-analysis-hw1-with-solution-2007-9-11 - Solutions to...

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Solutions to Homework 1 1. (10 points) Let Q be the set of all rationals in the interval [0 , 1] . Let S = { I 1 ,I 2 ,...,I m } be a finite collection of closed intervals covering Q. Show that m X k =1 v ( I k ) 1 . (0.1) On the other hand, for any ε> 0 , one can f nd S = { I 1 ,I 2 ,...,I m , ···} , which is a count- able collection of closed intervals covering Q, such that X k =1 v ( I k ) <ε. (0.2) In particular, (0.2) implies that | Q | e =0 . (Now you see the di f erence between the use of ”finite cover” and ”countable cover” .) Solution: For (0.1), we f rst assume that the intervals I 1 ,I 2 ,. . .,I m are nonoverlapping . In such case we clearly have (0.1). For arbitrary intervals I 1 ,I 2 ,. . .,I m with overlapping, one can throw away the over- lapping part and the remaining nonoverlapping part, which we denote it as J 1 ,J 2 ,. . .,J n , satis f es P n k =1 v ( J k
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This note was uploaded on 03/05/2009 for the course MATH 117 taught by Professor Akhmedov,a during the Spring '08 term at UCSB.

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real-analysis-hw1-with-solution-2007-9-11 - Solutions to...

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