Real numbers/Series/Rules/Fact/Proof/Exercise

Let

${\displaystyle \sum _{k=0}^{\infty }a_{k}{\text{ and }}\sum _{k=0}^{\infty }b_{k}}$

be convergent series of real numbers with sums ${\displaystyle {}s}$ and ${\displaystyle {}t}$. Prove the following statements.

1. The series ${\displaystyle {}\sum _{k=0}^{\infty }c_{k}}$ with ${\displaystyle {}c_{k}=a_{k}+b_{k}}$ is convergent with sum equal to ${\displaystyle {}s+t}$.
2. For ${\displaystyle {}r\in \mathbb {R} }$ the series ${\displaystyle {}\sum _{k=0}^{\infty }d_{k}}$ with ${\displaystyle {}d_{k}=ra_{k}}$ is also convergent with sum equal to ${\displaystyle {}rs}$.