Real analysis: cauchy/convergence

Let $\{b_n\}$ be a sequence of positive number such that $b_n \to 0$ and suppose that the terms in the sequence $\{a_n\}$ satisfy $|a_m−a_n| \leq b_n$ for all $m>n$. Prove that $\{a_n\}$ converges

i worked it out and till

we need to show that for any $\epsilon > 0$ there is some $n_0 \in \mathbb{N}$ such that
$|a_m-a_n| < \epsilon$ whenever $m,n > n_0$.

Assume $m \leq n$ then $k=n-m$
by using triangular law i reached till

$$|a_m -a_{m+k}|\leq|a_m-a_{m+1}|+|a_{m+1}-a_{m+2}|+....|a_{m+k-1} - a_{m+k}|$$