Basic Analysis I — Real Analysis · Problem 7.6.8

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · fixed-point-theorem

Problem

Suppose

f: X \to X

is a contraction for

k < 1

. Suppose you use the iteration procedure with

x_{n+1} \coloneqq f(x_n)

as in the proof of the fixed point theorem. Suppose

x

is the fixed point of

f

. a) Show that

d(x, x_n) \leq k^n d(x_1, x_0) \dfrac{1}{1-k}

for all

n \in \mathbb{N}

. b) Suppose

d(y_1, y_2) \leq 16

for all

y_1, y_2 \in X

, and

k = 1/2

. Find an

N

such that starting at any given point

x_0 \in X

d(x, x_n) \leq 2^{-16}

for all

n \geq N