hw3 - { s n } implies the convergence of {| s n |} . Is the...

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Homework 3 – Math 118A, Fall 2009 Due on Thursday, October 22nd, 2009 1. Let A and B be separated subsets of R k , suppose a A , b B , and deFne p ( t ) = (1 - t ) a + t b for t R . Put A 0 = p - 1 ( A ), B 0 = p - 1 ( B ). (a) Prove that A 0 and B 0 are separated subsets of R 1 . (b) Prove that there exists t 0 (0 , 1) such that p ( t 0 ) / A B . (c) Prove that every convex subset of R k is connected. 2. Prove that the convergence of
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Unformatted text preview: { s n } implies the convergence of {| s n |} . Is the converse true? 3. Calculate lim n ( n 2 + n-n ) 4. If s 1 = 2, and s n +1 = q 2 + s n , n = 1 , 2 , 3 , . . . , prove that { s n } converges, and that s n < 2 for n 1. 1...
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