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Unformatted text preview: Quiz 3 Solutions 1. (5 pts) Prove lim sup | s n | = 0 iff lim s n = 0. Solution: see homework 4 solutions. 2. Suppose that lim sup s n = 1. (a) (5 pts) Construct a subsequence ( s n k ) which converges to 1. Use only definitions, no theorems. Solution: Given any > 0, there exist infinitely many elements s n k in the interval (1- , 1+ ). (Make sure you can prove this statement. Its pretty easy to do by contradiction.) Choose s n = s . Then inductively choose s n k to be one of the infinitely many elements in the interval (1- 1 /k, 1+1 /k ), so that n k > n k- 1 (so we dont pick the same sequence element twice). Then because | s n k- 1 | < 1 /k for all n k , s n k 1. (Again, make sure you can prove this. It has to do with the fact that given any > 0, you can find a k so that 1 /k < .) Note: nobody got this right. Maybe I should put it on the final. (b) (5 pts) Is it true that s n 1? Prove or give a counterexample....
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