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3. Assume that f : R → R is such that |f(x) – f(y)| < A\x – y| for all x, y E R and some A E (0, 1). Pick xo E R arbitrarily, and construct a sequence (xn) as follows: xn+1 f(xn) for n > 0. (a) Prove that |æn+1 – Xn] < \|xn – xn-1| for all n > 1. (b) Prove that |æn+1– Xn[ < \"|x1 – xo] for all n > 1. (c) Prove that (xn) is convergent. (d) Let x* lim xn. Prove that f(x*) = x*. ||

3. Assume that f : R → R is such that |f(x) – f(y)| < A\x – y| for all x, y E R and some A E (0, 1). Pick xo E R arbitrarily, and construct a sequence (xn) as follows: xn+1 f(xn) for n > 0. (a) Prove that |æn+1 – Xn] < \|xn – xn-1| for all n > 1. (b) Prove that |æn+1– Xn[ < \"|x1 – xo] for all n > 1. (c) Prove that (xn) is convergent. (d) Let x* lim xn. Prove that f(x*) = x*. ||

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. Assume that f : R → R is such that |f(x) – f (y)| < A|x – y| for all x, y E R and some A E (0,1). Pick xo E R arbitrarily, and construct a sequence (xn) as follows: n+1 f (xn) for n > 0. (a) Prove that |Xn+1 – Xn[ < A|xn Xn-1| for all n > 1. (b) Prove that |xn+1 – Xn| < \"|x1 – xo] for all n > 1. (c) Prove that (xn) is convergent. (d) Let x* lim xn. Prove that f(x*) = x*.
Transcribed Image Text:3. Assume that f : R → R is such that |f(x) – f (y)| < A|x – y| for all x, y E R and some A E (0,1). Pick xo E R arbitrarily, and construct a sequence (xn) as follows: n+1 f (xn) for n > 0. (a) Prove that |Xn+1 – Xn[ < A|xn Xn-1| for all n > 1. (b) Prove that |xn+1 – Xn| < \"|x1 – xo] for all n > 1. (c) Prove that (xn) is convergent. (d) Let x* lim xn. Prove that f(x*) = x*.
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