Elementary proof, convergence of a linear combination of convergent series
Could you tell me how to prove that if two series $ \sum_{n=0} ^{\infty}x_n, \sum_{n=0} ^{\infty} y_n$ are convergent, then $\sum_{n=0} ^{\infty}(\alpha \cdot x_n + \beta \cdot y_n)$ is also convergent and its sum equals $\alpha \cdot \sum_{n=0} ^{\infty} x_n + \beta \cdot \sum_{n=0} ^{\infty} y_n$?
Solution 1:
Hint: Think about the definition of $\displaystyle \sum \limits_{n=0}^{+\infty}(a_n)\color{grey}{\left(=\lim \limits_{m\to+\infty}\left(\sum \limits_{n=0}^m(a_n)\right)\right)}$ and use $\lim$ properties.