$|x_{n + 1} - x_n| < \frac{1}{2^n} \Rightarrow (x_n)$ is Cauchy

Question

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?

Answer 1

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