Prove that $ (x_n)$ is convergent.

Question

Assume that 0 ≤ r < 1 and let $(x_n)$ be a sequence satisfying $|x_n − x_{n + 1}|$ ≤ $r^n$ $\forall n ∈ N$.

Prove that $(x_n)$ is convergent.

A sequence converges if it is monotonic and bounded above. I only do not know how to start.

How to prove this?

Answer 1

If you are allowed to use series and their convergence you can also proceed as follows:

• $x_n = x_0 + \sum_{k=1}^n (x_{k}-x_{k-1})$

But $s_n = \sum_{k=1}^n (x_{k}-x_{k-1})$ is (absolutely) convergent as

$$|\sum_{k=1}^n (x_{k}-x_{k-1})| \leq \sum_{k=1}^n |x_{k}-x_{k-1}|\leq \sum_{k=1}^n r^k = r\frac{1-r^n}{1-r}\stackrel{0\leq r <1}{\leq}\frac{r}{1-r}$$

As $(s_n)_{n \in \mathbb{N}}$ is convergent, it follows that $(x_n)_{n \in \mathbb{N}} = (x_0 + s_n)_{n \in \mathbb{N}}$ is convergent as well.

Answer 2

HINT

We have that

$$|x_n − x_{n + 1}|\to 0$$

then refer to the Cauchy criterion and to the related

• Please help prove that $(x_n)$ is a Cauchy sequence if $|x_{n+1} - x_n| \leq Cr^n$