hw4sol - Math 131A - Section 2 Spring 2010 Homework 4 All...

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Math 131A - Section 2 Spring 2010 Homework 4 All problems graded (out of 3 points for this short assignment). 10.1(b). bounded; (d) bounded; (f) bounded, non-increasing 10.2. Let ( s n ) be a bounded non-incerasing sequence. Let S = { s n | n N } and u = inf S , which is a real number since S is bounded. Let ± > 0. Then, u + ± is not a lower bound of S , so there exists N such that s N < u + ± . Then, since ( s n ) is non-increasing, we have that u s n s N < u + ± for all n > N , and so | s n - u | < ± for n > N . Therefore, lim n →∞ s n = u = inf S , which in particular shows ( s n ) converges. NOTES: This is almost exactly like the proof of Theorem 10.2 given in the textbook. Some care needed to be made in making the needed changes (make sure the inequalities all go in the correct direction!). 10.5. Let ( s n ) be an unbounded non-incerasing sequence. Then, since { s n | n N } is bounded above by s 1 , we must have that it is unbounded below. Let M < 0. Then, there exists N such that
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This note was uploaded on 11/20/2010 for the course MATH 131A 131A taught by Professor Kim during the Spring '10 term at UCLA.

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