Real series/Direct comparison test/Fact

Direct comparison test

Let ${}\sum _{k=0}^{\infty }b_{k}$ be a convergent series of real numbers and ${}{\left(a_{k}\right)}_{k\in \mathbb {N} }$ a sequence of real numbers fulfilling ${}\vert {a_{k}}\vert \leq b_{k}$ for all ${}k$.

Then the series

$\sum _{k=0}^{\infty }a_{k}$

is

absolutely convergent.

Proof, Write another proof