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Let f be a function defined on R satisfying \f(x) – f(y)| < 1z – yl, Vz, y E R. a. Show that f is a continuous function on R. b. Let æg = 1, a, = f(xn_1), n > 1. Use the Comparison Test to show that ("n – en-1) converges absolutely, and conclude that lim In exists. c. Let a, be the limit of (xn). Show that a, = f(x.).

Let f be a function defined on R satisfying \f(x) – f(y)| < 1z – yl, Vz, y E R. a. Show that f is a continuous function on R. b. Let æg = 1, a, = f(xn_1), n > 1. Use the Comparison Test to show that ("n – en-1) converges absolutely, and conclude that lim In exists. c. Let a, be the limit of (xn). Show that a, = f(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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Let f be a function defined on R satisfying |f(x) – f(y)| < -1z – y|, Vz, y E R. a. Show that f is a continuous function on R. b. Let zo = 1, z, = f(Tn_1), n > 1. Use the Comparison Test to show that (In – In-1) converges and conclude that lim In exists. c. Let r, be the limit of (zn). Show that a, = = f(x.).
Transcribed Image Text:Let f be a function defined on R satisfying |f(x) – f(y)| < -1z – y|, Vz, y E R. a. Show that f is a continuous function on R. b. Let zo = 1, z, = f(Tn_1), n > 1. Use the Comparison Test to show that (In – In-1) converges and conclude that lim In exists. c. Let r, be the limit of (zn). Show that a, = = f(x.).
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