131aw12hw2 - s , then s S . (a) Prove that, for any a,b R...

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Math 131A Problem Set #2 Due January 25 Exercises 7.4, 8.1(b)(c), 8.2(b)(e), 8.4, 8.7(a), 8.9, 8.10 from the textbook Additional Problem 1: (Uniqueness of Limits) We were a bit casual in lecture when speaking of the number L being the limit of the sequence ( s n ); by using the definite article the , we are implicitly assuming that there can only be one limit to a sequence. But how do we know that it’s not possible for a sequence to have two limits? Well, let’s prove that this is not possible: Prove that if a sequence ( s n ) converges to s and also converges to s 0 , then s = s 0 . Additional Problem 2: We say that S R is closed if whenever ( s n ) is a sequence from S and ( s n ) converges to a number
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Unformatted text preview: s , then s S . (a) Prove that, for any a,b R with a < b , that the interval [ a,b ] is closed. (b) Give two examples of an unbounded, closed set S R . Make sure to justify that your examples work. (c) Suppose that S R is a closed set that is bounded above. Prove that sup( S ) S . (d) A subset O R is said to be open if, for every x O , there is > 0 such that ( x-,x + ) O . For example, any interval ( a,b ) is open. (You dont need to prove this; just convince yourself that this is true.) Prove, for a set S R , that S is closed if and only if R \ S is open....
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This note was uploaded on 01/30/2012 for the course MATH 131a taught by Professor Hitrik during the Spring '08 term at UCLA.

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