Geometric Mean Problem

This week, I wanted to try to solve on my own one of the problems we didn't go over the solution for. This is from a couple of weeks ago:

"Prove that for every pair of non-negative real numbers (x,y), if x is greater than y, then the geometric mean, sqrt(xy) is less than the arithmetic mean, (x + y)/2."

y < x
0 < (x - y)
0^2 < (x - y)^2
0 < x^2 - 2xy + y^2
4xy < x^2 + 2xy + y^2
xy < (x+y)^2 / 4
sqrt(xy) < (x + y)/2

I'm not comfortable enough yet with proof structures, but there's the math! This sort of algebraic problem-solving I find really fun.