If $\sum_{n=1}^\infty\vert a_{n+k}-a_n\vert<\infty$, then $\{a_n\}_{n=1}^\infty$ has a convergent subsequence

Let $M= \sum\limits_{n=1}^{\infty} |a_{n+k}-a_n|$. Check that $ \sum\limits_{n=1}^{\infty} |a_{k(n+1)}-a_{kn}| \leq M$. The sequence $b_n=a_{k n}$ satisfies $\sum |b_{n+1}-b_n| <\infty$. This implies that $(b_n)$ is Cauchy.