Lebl-Basic-Analysis-I · Problem 2.4.2

Sequences and Series

Mathematics · cauchy-sequences

Problem

Let {xn}n=1∞\{x_n\}_{n=1}^\infty be a sequence such that there exists a positive C<1C < 1 and for all nn, ∣xn+1−xn∣≤C∣xn−xn−1∣.|x_{n+1} - x_n| \leq C|x_n - x_{n-1}|. Prove that {xn}n=1∞\{x_n\}_{n=1}^\infty is Cauchy. Hint: You can freely use the formula (for C≠1C \neq 1) 1+C+C2+⋯+Cn=1−Cn+11−C.1 + C + C^2 + \cdots + C^n = \frac{1 - C^{n+1}}{1 - C}.

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