Use the Cauchy–Schwarz inequality:
Equality obtains if and only if the vectors and in are linearly dependent; i.e. if and only if . Since , this won’t happen and so the inequality is always strict.
Add them up,
Cancel 2 on both sides to wrap it up
Yes I am aware that you can get it by a simple application of CS. But when we were in school we were taught only AM-GM. So I thought probably the poster wanted the proof using that