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Unformatted text preview: segments shown. 1-7 1-6 1-5 1-4 1-3 1-2 1 1 3. Problem 14 I. Let X = ( x n ) be a sequence of strictly positive real numbers such that lim ( x n +1 /x n ) < 1. Show that for some r with 0 < r < 1 and some C > 0, then we have < x n < Cr n for all suciently large n N . Use this to show that lim ( x n ) = 0. 4. Let ( x n ) be a sequence of numbers in R p . Suppose that there is an x R p such that every subsequence ( y n ) of ( x n ) has a subsequence ( z n ) of ( y n ) such that lim z n = x . Show that lim ( x n ) = x . Hint: Proof by contradiction. What does it mean to say that x n does not converge to x ? 1...
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- Summer '07