Intro to Analysis in-class_Part_18

# Intro to Analysis in-class_Part_18 - 3.1 Limits and bounds...

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3.1 Limits and bounds 41 Proof. Use the Cauchy sequence construction; see Strichartz. Theorem 3.1.7. A monotone increasing sequence { a n } ⊆ R is convergent iﬀ it is bounded above. In this case, the limit is the sup of the set { a 1 ,a 2 ,... } . Proof. ( ) If it’s convergent, then it’s Cauchy. If it’s Cauchy, then it’s bounded. ( ) Since R is l-u-b and the set sup { a 1 ,a 2 ,... } is bounded above, let α := sup { a 1 ,a 2 ,... } . Then α is an upper bound, so a k α, k . Then α - 1 m is not an upper bound for the sequence, for any m N i.e., there is some N for which α - 1 m < a N . Since the sequence is monotone, this will also be true for every term thereafter: j J = α - 1 m < a n . By the Squeeze Thm, lim( α - 1 m ) = α implies that lim a n = α . Example 3.1.2. We use two results from discrete math. Binom formula: (1 + x ) k = 1 + kx + ··· + ± k i x i + ··· + x n Geometric sum (ﬁnite): 1 + r + r 2 + ··· + r n = 1 - r n +1 1 - r . When

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## This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis in-class_Part_18 - 3.1 Limits and bounds...

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