Basic Analysis I — Real Analysis · Problem 7.6.9

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · fixed-point-theorem

Problem

Let

f(x) \coloneqq x - \dfrac{x^2 - 2}{2x}

(you may recognize Newton's method for

\sqrt{2}

). a) Prove

f\bigl([1,\infty)\bigr) \subset [1,\infty)

. b) Prove that

f: [1,\infty) \to [1,\infty)

is a contraction. c) Show that the fixed point theorem applies, find the unique

x \geq 1

such that

f(x) = x

, and show that

x = \sqrt{2}

. Note: In particular, the technique from the proof of the theorem can be used to approximate

\sqrt{2}