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

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Balanced signs

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