TITLE: $|x_{n + 1} - x_n| < \frac{1}{2^n} \Rightarrow (x_n)$ is Cauchy
QUESTION [5 upvotes]: Let $(x_n)$ be a real sequence with the property that for all $n \in \mathbb{N}$, 
$$|x_{n + 1} - x_n| < \frac{1}{2^n}$$
I want to show, using the definition of a Cauchy sequence, that $(x_n)$ must be Cauchy.
I have found that the property implies that for any $(m, n) \in \mathbb{R}^2$, assuming without loss of generality that $m > n$, it must be true that
$$|x_n - x_m| \leq \sum\limits_{i = n}^m \frac{1}{2^i}$$
How can I proceed from there ? Is this even the right way to approach this problem?

REPLY [4 votes]: Here is a slightly different argument which is not presented on the linked duplicates:

$s_n = \sum_{k=1}^{n}(x_k - x_{k-1}) = x_n - x_0$ is (absolutely) convergent
$\Rightarrow x_n = s_n + x_0$ is convergent
convergent sequences are Cauchy-sequences