Let $\{X_n\}$ be a sequence and let $r\in \mathbb{R}$ such that $0<r<1$. Suppose that $\left|X_{n+1}-X_{n}\right|\le r|X_{n}-X_{n-1}|\ \forall n>1$

Question

Prove that $\{X_n\}$ is a Cauchy sequence and hence convergent.

edit: Thanks to the user that fixed the syntax! That was pretty awesome of you!

Answer 1

We assume that $\{X_n\}$ is a real sequence. Letting $X_{n+1}-X_n=a_n$, we see that $|a_n|\le r^n|a_0|\implies$ for $m>n$, $|X_m-X_n|\le \sum_{k=1}^{m-n}|X_{n+k}-X_{n+k-1}|\le \sum_{k=1}^{m-n}r^{n+k-1}|a_0|=r^n\frac{1-r^{m-n}}{1-r}|a_0|\le A r^n $ where $A=\frac{1}{1-r}$. So, for a given $\epsilon >0$, if we choose $N$ such that $r^N A<\epsilon$, then, we have $\forall\ m>n\ge N$, $|X_m-X_n|\le Ar^n\le Ar^N<\epsilon$, hence the sequence is Cauchy.

Answer 2

Hint

1. $|x_n-x_{n-1}| \leq r^{n-1}|x_2-x_1|$.

2. $|(a-b)-(b-c)| \leq |a-b|+|b-c|$

Can you deal with it now?

Answer 3

Use that $|x_{n+p}-x_n|\leq|x_{n+p}-x_{n+p-1}|+...+|x_{n+1}-x_n|$, for every $p\in \mathbb N$ and the hypothesis.