6:19 PM
Well, how 'bout this? $$x_1y_1 + x_2y_2 \le 2\sqrt{x_1y_1x_2y_2} = \sqrt{2x_1x_2}\sqrt{2y_1y_2}\le\sqrt{x_1^2+x_2^2}\sqrt{y_1^2+y_2^2.$$
But I don't see that this generalizes to $n$, since then we get $n$th powers.
Ugh. Left off the final }.
$$x_1y_1 + x_2y_2 \le 2\sqrt{x_1y_1x_2y_2} = \sqrt{2x_1x_2}\sqrt{2y_1y_2}\le\sqrt{x_1^2+x_2^2}\sqrt{y_1^2+y_2^2}.$$
