Let f be a function defined on R satisfying |f(x) – f(y)| < ;|æ – y), Væ,y € R. a. Show that f is a continuous function on R. b. Let xo = 1, xn = f(xn-1), n > 1. Use the Comparison Test to show that 00 > (*n – xn-1) n=1 converges absolutely, and conclude that lim an exists. c. Let x, be the limit of (xn). Show that x, = = f(x.).
Let f be a function defined on R satisfying |f(x) – f(y)| < ;|æ – y), Væ,y € R. a. Show that f is a continuous function on R. b. Let xo = 1, xn = f(xn-1), n > 1. Use the Comparison Test to show that 00 > (*n – xn-1) n=1 converges absolutely, and conclude that lim an exists. c. Let x, be the limit of (xn). Show that x, = = f(x.).
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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Transcribed Image Text:Let f be a function defined on R satisfying
|f(x) – f(y)| < la – yl, Væ, y E R.
a. Show that f is a continuous function on R.
b. Let xo = 1, xn = f(xn-1), n >1. Use the Comparison Test to show that
00
(*n – xn-1)
n=1
converges absolutely, and conclude that
lim xn
exists.
c. Let x, be the limit of (xn). Show that x, =
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