Simple Inequality Proof

Question

Prove the following inequality

$\ xX + yY \leq \sqrt{ x^2 + y^2}\cdot\sqrt {X^2 + Y^2}$

where $x, y, X$ and $Y$ are real numbers.

Answer 1

Use Cauchy-Schwarz Inequality for $\mathbb{R}^2$ with the "vectors" $(x,y)$ and $(X,Y)$.

Answer 2

(1) Make square both sides

(2) Use the fact that $$2AB \leq A^2+ B^2$$