Abel's Inequality

Let $\{f_n\}$ and $\{a_n\}$ be Sequences with $f_n \geq f_{n+1} > 0$ for $n=1$, 2, ..., then

$$\left\vert{\sum_{n=1}^m a_n f_n}\right\vert \leq A f_1,$$

where

$$A = \max\{\vert a_1\vert,\left\vert{a_1+a_2}\right\vert,\ldots,\left\vert{a_1+a_2+\ldots+a_m}\right\vert\}.$$