Suppose $(a_n)$ is a non-increasing sequence of positive real numbers and $\varepsilon_i = \{\pm 1\},\ \forall i \in \mathbb{N}$ such that $\sum\limits_{i=1}^\infty \varepsilon_i a_i$ is convergent.
Is it true that $\lim\limits_{n\to \infty}(\varepsilon_1+\varepsilon_2+...+\varepsilon_n) a_n=0$?