Proposition: $|x+y| \leq |x| + |y|$

Proof : $(x+y)^2 = x^2 + 2xy + y^2 $

$\leq |x|^2 + 2|x||y| + |y|^2$ 

$= (|x| + |y|)^2$

So $(x+y)^2 \leq (|x|+|y|)^2 \Rightarrow \sqrt{(x+y)^2} \leq \sqrt{(|x| + |y|)^2} \Rightarrow |x+y| \leq |x| + |y|$

Here the fact was used that $\sqrt{(x)^2} = |x|$

I understand what is going on in this proof. No questions in that regard. I have known about this proof for awhile and I never questioned it before. I was randomly thinking about it and started overthinking the situation most likely but maybe someone can shed light on my concerns. 

I feel like $x^2 + 2xy + y^2 \leq |x|^2 + 2|x||y| + |y|^2$ is just hand waved like it is well known and maybe it is. Is something like $x \leq |x|$ just assumed or does it actually need to be proven? Sorry if this question is nonsense, but I majored in math so I always think of random things like this!