Basic Analysis I — Real Analysis · Problem 2.2.9

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · convergence-tests

Problem

Suppose

\{x_n\}_{n=1}^\infty

is a sequence,

x \in \mathbb{R}

, and

x_n \neq x

for all

n \in \mathbb{N}

. Suppose the limit

L := \lim_{n\to\infty} \frac{|x_{n+1} - x|}{|x_n - x|}

exists and

L < 1

. Show that

\{x_n\}_{n=1}^\infty

converges to

x