Question

Let I be a closed interval and let 0 < a < 1. Let f: I ? I satisfy the inequality |f(x) - f(y)| ? a|x - y| for each x ? I and y ? I. (Such a function is called a contraction mapping.) Prove that f is continuous on I. Let x1 ? I and define xn+1 = f(xn) (n = 1, 2, …). Prove that the sequence ?xn? converges and that its limit l satisfies l = f(l). [Hint: exercise 5.21(1).]

          Let I be a closed interval and let 0 < a < 1. Let f: I ? I satisfy the inequality
|f(x) - f(y)| ? a|x - y|
for each x ? I and y ? I. (Such a function is called a contraction mapping.) Prove that f is continuous on I.
Let x1 ? I and define xn+1 = f(xn) (n = 1, 2, …). Prove that the sequence ?xn? converges and that its limit l satisfies l = f(l). [Hint: exercise 5.21(1).]

Let I be a closed interval and let 0 < a < 1. Let f: I ? I satisfy the inequality
|f(x) - f(y)| ? a|x - y|
for each x ? I and y ? I. (Such a function is called a contraction mapping.) Prove that f is continuous on I.
Let x1 ? I and define xn+1 = f(xn) (n = 1, 2, …). Prove that the sequence ?xn? converges and that its limit l satisfies l = f(l). [Hint: exercise 5.21(1).]

Added by John B.

Calculus: Early Transcendentals

James Stewart 8th Edition

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07/13/2022

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Let I be a closed interval and let 0 < a < 1. Let f: I → I satisfy the inequality |f(x) - f(y)| ≤ a|x - y| for each x ∈ I and y ∈ I. (Such a function is called a contraction mapping.) Prove that f is continuous on I. Let x1 ∈ I and define xn+1 = f(xn) (n = 1, 2, …). Prove that the sequence ⟨xn⟩ converges and that its limit l satisfies l = f(l). [Hint: exercise 5.21(1).]