# Balanced signs

Assume $(p_n)$ is a nonincreasing sequence of positive integers whose sum $p_1+p_2+\cdots$ diverges and assume signs $(\varepsilon_n)$ are chosen so that the modified series $\varepsilon_1p_1+\varepsilon_2p_2+\cdots$ converges. Then

$\liminf\limits_{n\to\infty}\dfrac{\varepsilon_1+\cdots+\varepsilon_n}n \leqslant 0\leqslant \limsup\limits_{n\to\infty}\dfrac{\varepsilon_1+\cdots+\varepsilon_n}n$

Assume now only the signed series converges, with no hypothesis on the unsigned sequence. Show

$\lim\limits_{n\to\infty} (\varepsilon_1+\cdots+\varepsilon_n)p_n=0$