Lebl-Basic-Analysis-I · Problem 2.2.9

Sequences and Series

Mathematics · facts-about-limits-of-sequences

Problem

Suppose {xn}n=1∞\{x_n\}_{n=1}^\infty is a sequence, x∈Rx \in \mathbb{R}, and xn≠xx_n \neq x for all n∈Nn \in \mathbb{N}. Suppose the limit L:=lim⁡n→∞∣xn+1−x∣∣xn−x∣L := \lim_{n\to\infty} \frac{|x_{n+1} - x|}{|x_n - x|} exists and L<1L < 1. Show that {xn}n=1∞\{x_n\}_{n=1}^\infty converges to xx.

Related theory

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