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Assume f: R→→ R. and for some c < 1 we have for all x, y € R, |f(x) = f(y)| ≤ c|x − y|. (a) Prove that f is uniformly continuous. (b) Pick and ₁ ER and inductively define a sequence by letting In = f(xn-1) (for n ≥ 2). Prove that the sequence (n) is Cauchy. (c) If (n) is the sequence from (b), and (n) → a, then f(a) = a (so a is a fixed point of f). (d) Prove that a the only fixed point of f.

Assume f: R→→ R. and for some c < 1 we have for all x, y € R, |f(x) = f(y)| ≤ c|x − y|. (a) Prove that f is uniformly continuous. (b) Pick and ₁ ER and inductively define a sequence by letting In = f(xn-1) (for n ≥ 2). Prove that the sequence (n) is Cauchy. (c) If (n) is the sequence from (b), and (n) → a, then f(a) = a (so a is a fixed point of f). (d) Prove that a the only fixed point of f.

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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6. Assume f: R → R and for some c < 1 we have for all x, y € R, |f(x) = f(y)| ≤ c|x − y\. (a) Prove that f is uniformly continuous. (b) Pick and ₁ R and inductively define a sequence by letting In = f(n-1) (for n ≥ 2). Prove that the sequence (n) is Cauchy. (c) If (x) is the sequence from (b), and (xn) → a, then f(a) = a (so a is a fixed point of f). (d) Prove that a the only fixed point of f.
Transcribed Image Text:6. Assume f: R → R and for some c < 1 we have for all x, y € R, |f(x) = f(y)| ≤ c|x − y\. (a) Prove that f is uniformly continuous. (b) Pick and ₁ R and inductively define a sequence by letting In = f(n-1) (for n ≥ 2). Prove that the sequence (n) is Cauchy. (c) If (x) is the sequence from (b), and (xn) → a, then f(a) = a (so a is a fixed point of f). (d) Prove that a the only fixed point of f.
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