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Unformatted text preview: Math 142A (P. Fitzsimmons) First Midterm Solutions October 17, 2001 1. (a) A sequence { a n } is monotone provided it is increasing ( i.e. , a n +1 ≥ a n for all n ) or decreasing ( i.e. , a n +1 ≤ a n for all n ). (b) A sequence { a n } is bounded provided there are numbers L and U such that L ≤ a n ≤ U for all n . Thus, a sequence is bounded if it is both bounded above and bounded below. (c) The completeness property of the real number system is this: If an increasing sequence is bounded above, then it converges. 2. Use the definition of limit (of a sequence) to show that lim n →∞ 2 n 2 2 + n 2 = 2 . Solution. We begin by estimating the difference between the n th term of the sequence and its limit: ( † ) vextendsingle vextendsingle vextendsingle vextendsingle 2 n 2 2 + n 2 2 vextendsingle vextendsingle vextendsingle vextendsingle = vextendsingle vextendsingle vextendsingle vextendsingle 4 2 + n 2 vextendsingle vextendsingle vextendsingle vextendsingle = 4 2 + n 2 ≤...
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 Fall '01
 Fitzsimmons
 Math, Calculus, Mathematical analysis, Limit of a sequence, Cauchy sequence, P. Fitzsimmons

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