4. Let f be differentiable on R with a = sup{|ƒ'(x)| : x € R} < 1.f(sn-1) for n ≥ 1. Prove that(i) Pick a number so € R and inductively define sn={n} is a convergent sequence in R.(ii) Show that f has a fixed point, that is, f(s) = s for some s E R.

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Answered: 4. Let f be differentiable on R with a…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 75E

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4. Let f be differentiable on R with a = sup{|f'(x)| : x € R} < 1. (i) Pick a number so € R and inductively define sn = f(sn-1) for n ≥ 1. Prove that {n} is a convergent sequence in R. (ii) Show that f has a fixed point, that is, f(s) = s for some s € R.