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hom2sol - Solutions to Homework 2 February 2 2009 18.4 Let...

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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 0 / 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 0 . With the hint and this property, a good choice will be f ( x ) = 1 x - x 0 Also note, it’s good practice to show that the f ( x ) chosen is continuous everywhere but at x 0 , but it wasn’t marked wrong if you didn’t prove it. Proof: Let f ( x ) = 1 x - x 0 , and the sequence ( x n ) be in S and converge to x 0 . Since f ( x ) is continuous on R x 0 so it is continuous on S . lim n →∞ f ( x n ) = lim n →∞ 1 x n - x 0 . Note that for arbitrarily small ǫ > 0 N > 0 s.t. | x n x 0 | < ǫ which implies 1 ǫ < 1 | x n - x 0 | . Thus for arbitrarily large M = 1 there exist an N > 0 (namely the same one above) s.t. | f ( x n ) | > M thus f ( x n ) diverges to infinity as x n x 0 , meaning f ( x ) is unbounded in S .
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