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Unformatted text preview: Solutions to Homework 2 February 2, 2009 18.4: Let S R and suppose a sequence ( x n ) in S that converges to a number x / S . Show that there an unbounded continuous function on S . Note: In other words, we want to show that there is a function f defined on S (and thus defined on ( x n )) such that it is unbounded as x approaches x . With the hint and this property, a good choice will be f ( x ) = 1 x- x Also note, its good practice to show that the f ( x ) chosen is continuous everywhere but at x , but it wasnt marked wrong if you didnt prove it. Proof: Let f ( x ) = 1 x- x , and the sequence ( x n ) be in S and converge to x . Since f ( x ) is continuous on R x so it is continuous on S . lim n f ( x n ) = lim n 1 x n- x . Note that for arbitrarily small > N > 0 s.t. | x n x | < which implies 1 < 1 | x n- x | . Thus for arbitrarily large M = 1 / there exist an N > 0 (namely the same one above) s.t. | f ( x n ) | > M thus f (...
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This document was uploaded on 10/01/2009.
- Spring '09