Proof that this sequence is Cauchy

Question

How to prove that if ${a_n}$ is a Cauchy sequence, if $x_n$ is a sequence and there is a constant $C$ such that,

$$|x_n-x_m| \le C|a_n-a_m|$$

Then $x_n$ is also a Cauchy sequence. Any hints or solutions would be appreciated.

Answer 1

So if $(a_n)$ is Cauchy, we know that for every $\varepsilon > 0$ there is an $N$ such that for all $n,m>N$, $|a_n-a_m| < \frac{\varepsilon}{C}$. What happens if you use this in conjunction with your inequality...?