Unformatted text preview: 1 and  x nx n +1  ≤ k  x n1x n  for all n ∈ N . In this exercise I will guide you through a proof that all contractive sequences converge. (a) Prove by induction the sequence will satisfy the inequality  x nx n +1  ≤ k n1  x 1x 2  for all n ≥ 1. Then use this to rewrite  x nx m  for arbitrary n,m in terms of  x 1x 2  . (b) (9.18(a)) Prove 1 + k + k 2 + ··· + k n = 1k n +1 1k by induction. (c) Prove that k n → 0. (d) Combine parts (b) and (c) to prove that ∀ ± > 0, ∃ N so that ∀ m > n > N , k n + k n +1 + ··· + k m < ε . (e) Combine (a)(d) to prove that all contractive sequences are Cauchy, hence convergent. 1...
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 Winter '11
 Wiley
 Differential Equations, Equations

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