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- Jun 20, 2014

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Consider a sequence of real numbers $(x_n)_{n = 1}^\infty$ such that $\sum\limits_{n = 1}^\infty \lvert x_n y_n\rvert$ converges for every real sequence $(y_n)_{n = 1}^\infty$ such that $\sum\limits_{n = 1}^\infty y_n^2$ converges. Prove that $\sum\limits_{n = 1}^\infty x_n^2$ converges.

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