Prove that the sums and differences of convergent sequences are convergent
This is the trick to the proof.
$\displaystyle \left| {\left( {a + b} \right) - \left( {a_n + b_n } \right)} \right| = \left| {\left( {a - a_n } \right) + \left( {b - b_n } \right)} \right| \le \left| {\left( {a - a_n } \right)} \right| + \left| {\left( {b - b_n } \right)} \right|
$