TITLE: Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$
QUESTION [8 upvotes]: How can I prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$
I have tried substitute $x,y$ for numbers, which turns out right, but I don't know how to reason here.
Thanks in advance!

REPLY [5 votes]: Just for the record, a "geometric" proof:

OBS: Which provides a nice idea that can be generalized to prove the Young's Inequality.