Simple question about inner product spaces

Question

$$|\langle x,y\rangle|^2\leq\langle x,x\rangle\langle y,y\rangle$$ Is true in any inner product space, please if someone can show me how to prove the next statement out of the first one $$\sqrt{\langle x+y,x+y \rangle}\le \sqrt{\langle x,x \rangle} + \sqrt{\langle y,y \rangle}$$

I hope I've been clear enough, Thanks.

Answer 1

Square both sides. Note that $$ \langle x+y,x+y \rangle = \langle x,x+y \rangle + \langle y,x+y \rangle =\\ \langle x,x \rangle + \langle x,y \rangle + \langle y,x \rangle + \langle y,y \rangle $$ And that $$ \langle x,y \rangle + \langle y,x \rangle = 2 \text{Re} (\langle x,y \rangle) \leq 2\sqrt{\langle x,x \rangle \langle y,y\rangle} $$