Let {bn} be a bounded sequence of nonnegative numbers and r be any number such that 0≤r<1.
sn = b1r + b2r^2 + ….. + bnr^n for every index n.
Use the monotone convergence theorem to prove that the series {sn} converges.
Because all these numbers are nonnegative, then it should be clear to you that $\displaystyle \left( {s_n } \right)$ is an non-decreasing sequence. Now let B be the bound such that $\displaystyle \left( {\forall n} \right)\left( {\left| {b_n } \right| \le B} \right)$.
It should also be clear that, $\displaystyle s_n = \sum\limits_{k = 1}^n {b_k r^k } \le B\sum\limits_{k = 1}^n {r^k } \le \frac{{Br}}{{1 - r}}$.
What do you know about a bounded non-decreasing sequence?