Linear Combination of Convergent Series

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

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

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

Proof
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