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quiz3mat25

# quiz3mat25 - Quiz 3 Solutions 1(5 pts Prove lim sup |sn | =...

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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. It’s pretty easy to do by contradiction.) Choose s n 0 = s 0 . 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 don’t 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. Solution: No, this is not true. For a counterexample, consider the se- quence defined by s n = ( - 1) n . 1

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3. (10 pts) True or false. 2 pts each: 1 pt for correct answer, 1 pt for justification. (a) The set of subsequential limits of a bounded sequence is always non-empty.
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