Question

Prove that the sum of two convergent sequence is convergent.

Answer 1

Let A and B be the points of convergence of the two respective series. The convergence of the two series implies that given any $\varepsilon>0$ , there exists an integer $N_{0}$ such that for any $N \geq N_{0}$ , we have

$\begin{aligned} &\left|A-\sum_{n=1}^{N} a_{n}\right|<\varepsilon \\ &\left|B-\sum_{n=1}^{N} b_{n}\right|<\varepsilon \end{aligned}$

We claim that $\sum_{n=1}^{\infty} \frac{a_{n}+b_{n}}{2}$ converges to $\frac{A+B}{2}$ . Indeed, for all $N \geq N_{0}$ , we can use the triangle inequality to get

$\left|\frac{A+B}{2}-\sum_{n=1}^{N} \frac{a_{n}+b_{n}}{2}\right| \leq \frac{1}{2}\left|A-\sum_{n=1}^{N} a_{n}\right|+\frac{1}{2}\left|B-\sum_{n=1}^{N} b_{n}\right|<\varepsilon$

Hence, the sum of two convergent sequence is convergent.