# In general, is the sum of a product less than or equal to the product of its sums?

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 $$n=\infty$$?

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 $$0$$