Basic Analysis I — Real Analysis · Problem 7.6.5

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · fixed-point-theorem

Problem

Let

f: [0, 1/4] \to [0, 1/4]

be defined by

f(x) \coloneqq x^2

. a) Show that

f

is a contraction, and find the best (smallest)

k

from the definition that works. b) Find the fixed point and show directly that it is unique.