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Is there any sequence $a_n$ of nonnegative numbers for which $\sum_{n \geq 1}a_n^2 <\infty$ ...
Solution (based on Buczolich Zoltán's solution): There exists such a sequence. For every $M \in \mathbb{N}$, choose a set $\mathcal{P}_M = \left\{ p_{j, M} : j = 1, \ldots, l_M \right\}$ of distinct …