Linear Combination of Convergent Series

Theorem
Let $\left\langle{a_n}\right\rangle_{n \mathop \ge 1}$ and $\left\langle{b_n}\right\rangle_{n \mathop \ge 1}$ be sequences of real numbers.

Let the two series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ converge to $\alpha$ and $\beta$ respectively.

Let $\lambda, \mu \in \R$ be real numbers.

Then the series $\displaystyle \sum_{n \mathop = 1}^\infty \left({\lambda a_n + \mu b_n}\right)$ converges to $\lambda \alpha + \mu \beta$.