real analysis - Two sequences, one of them bounds difference of the other and converges to $0$. Show that the other sequence converges.

That is, let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be sequences such that $b_n \overset{n \to \infty}{\to} 0$ and for all $k \in \mathbb{N}$ and $l \geq k$,
$$|a_l - a_k| < b_k\text{.}$$
Show that $(a_n)_{n=1}^{\infty}$ is Cauchy.

My guess is that $a_n \to 0$. So, we could try working from the definition of convergence and show that $a_n \to 0$, but it isn't clear to me how to show that $|a_n| < \epsilon$.

Hints, not complete solutions, are appreciated.

Edit: Could the squeeze theorem potentially be useful here?

Answer

Hint: Let $\epsilon>0.$ Then there exists $N$ such that $b_n <\epsilon$ for $n\ge N.$ What happens if $k,l\ge N?$