Suppose $|a_n| \leq b_n - b_{n+1}$ where $b_n$ decreases monotonically to zero. Prove that $\sum_{n=1}^{\infty} a_n$ converges absolutely.
My thoughts were $\sum_{n=1}^{\infty} |a_n| \leq \sum_{n=1}^{\infty} (b_n - b_{n+1})$.
Now I'd to use the fact $b_n$ is monotonically decreasing to zero to show the sum on the right converges so by comparison so must the sum on the left, which proves what I want. My trouble is in formally proving that the right sum converges.
Informally I know $b_n \geq b_{n+1} \geq b_{n+2} ...$ and eventually for some $k$, $b_k = 0$, so we just have a finite number of terms we're summing up so it converges to whatever that is. I don't know if I can take for granted that the sequence hits $0$ at a finite $k$. Even then the details seem a bit hazy.