Assume that ($a_n$) converges to a and ($b_n$) converges to b. Prove that ($a_n-b_n$) converges to $a-b$.

Question

I used a similar mean to the one that solves $a_n+b_n$ converges to a+b. However, I found it to be difficult to solve $a_n-b_n$ = $a-b$, since the reverse triangle inequality only suggests $\mid a_n-a\mid - \mid b_n - b \mid \; < \; \mid a_n-a-b_n+b \mid \; = \; \mid (a_n-b_n)-(a-b) \mid$. However, $\mid a_n-a\mid - \mid b_n - b \mid \; <\; \epsilon$ does not necessarily indicate $\mid (a_n-b_n)-(a-b) \mid \; < \; \epsilon$. What can I do?