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Let $(a_n)_{n\in\mathbb{N}}$ be a sequence such that lim inf $|a_n|=0$. Prove that there is a subsequence $(a_{n_{k}})_{k\in\mathbb{N}}$ such that $\sum_{k=1}^\infty a_{n_k}$ converges.

I'm thought about showing that you can make a subsequence that is smaller then $1/n^2$. Is this the right strategy ? Any hints ? Should I use the Cauchy criterion ?

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Yes, that would work. You mean of course "find a subsequence $(a_{n_k})$ such that $|a_{n_k}|\le 1/k^2$ for all $k$". –  David Mitra Feb 11 '13 at 13:30

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up vote 4 down vote accepted

Show that for each $i\in N \exists n_i$ (of course $n_i<n_{i+1}$)such that $|a_{n_i}|<1/2^i$. Or even $1/i^2$ will work. The existence of such a subsequence easily follows from the definition of lim inf.

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