**Lemma**: If $\sum_{n=1}^{\infty}= A$ and $\sum_{n=1}^{\infty} b_{n} = B$ then $\sum_{n=1}^{\infty} (a_{n}+b_{n})= A+B$ **Proof**: We have$\sum_{n=1}^{k}a_{n} \to A \quad \text{ and } \quad \sum_{n=1}^{\infty} b_{n} \to B$so by [[Algebra of Limits of Convergent Sequences|algebra of limits]]$\sum_{n=1}^{k}(a_{n} + b_{n})= \sum_{n=1}^ {k} a_{n} +\sum_{n=1}^{k} b_{n} \to A+B$