This is the arithmetic-mean geometric mean inequality.

Let x>=0 and y>=0.

Then,

0<=(x-y)^2 because a square is always non-negative.

Expand,

0<=x^2-2xy+y^2

Add 4xy to both sides,

4xy<=x^2+2xy+y^2=(x+y)^2

Take square roots since both are non-negative,

sqrt(4xy)<=sqrt(x+y)^2

2sqrt(xy)<=|x+y|=x+y

Divide by 2,

sqrt(xy)<=(x+y)/2