real_analysisii_homework4

real_analysisii_homework4 - Homework 4 H ector Guillermo Cu...

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Unformatted text preview: Homework 4 H ector Guillermo Cu ellar R os February 21, 2006 6.2 (a) If s n 0, then for every > 0 there exist N R such that n > N implies s n < . True. According to Definition 16.2, if s n 0, then fore each > 0 there exists a real number N such that for all n N , n > N implies that | s n- | < , then | s n | < , clearly s n < . (b) If for every > 0 there exists N R such that n > N impies s n < , then s n 0. False, the sequence s n =- n is always less than and it does not converge to any number. (c) Given sequences ( s n ) and ( a n ), if for some s R , k > 0 and m N we have | s n- s | k | a n | for all n > m , then lim s n = s . False. Consider s n = (- 1) n , a n = n k and s = 0, none of the sequences converge, furthermore | s n- | 1 k n k = n for all n . (d) If s n t and s n t , then s = t . True by Theorem 16.14....
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This note was uploaded on 03/05/2009 for the course MATH 117 taught by Professor Akhmedov,a during the Spring '08 term at UCSB.

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real_analysisii_homework4 - Homework 4 H ector Guillermo Cu...

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