I'm given $\{b_n\}$ is a bounded set of non-negative numbers and $r$ such that $0 \leq r < 1$. I need to show the sequence $\{s_n\} = b_1r +b_2r^2 + ... +b_nr^n $ converges. Ideally with the monotone convergence theorem.
So starting out I know $\{b_n\}$ is a bounded set. So $|s_n| \leq Mr + Mr^2 + ... + Mr^n$ and $Mr + Mr^2 + ... + Mr^n = M(r+r^2 + ... + r^n) = M(\frac{1}{1-r})$ so $|s_n| \leq M\frac{1}{1-r}$ giving me that $s_n$ is bounded. I also know that $r,r^2,r^3,...,r^n$ is decreasing monotonically since $0 \leq r < 1$
But at this point I get stuck. I don't know anything about $\{b_n\}$ except that it is bounded and non-negative. Can anybody give me a hint as to where I should be going from here? I know I need to show that somehow $\{s_n\}$ is monotone increasing since we keep adding to it in smaller and smaller amounts. But I don't know how I can show this with the information I have.
Any hints / pointers would be much appreciated.
Also I would ask that people refrain from posting a full proof. I'm trying to learn this material.