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I was looking at this answer trying to understand more about the relation of sums, and it seems to suggest that

$$ \sum_{k=1}^{n}x_ky_k\leq\sum_{k=1}^{n}x_k\sum_{k=1}^{n}y_k $$

Is this true in general, and in general if $n=\infty$?

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Are the $x_k$ and $y_k$ all non-negative? If so, the answer is obviously yes by expanding the right hand product, as shown in the red and blue chart in an answer to the linked question; the right hand side has every term on the left hand side plus many more non-negative terms

If some of the terms can be negative, consider $x_1=y_1=+1$ and $x_2=y_2=-1$, making the left hand side $2$ and the right hand side $0$

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