Where Does the Less Than Symbol Disappear in the Triangle Inequality Proof?

[Punkyc7]

Im curios as to why the inquality is
||x+y||$$\leq$$||X||+||y||

but the end of the proof is

=(||x||+||Y||)^2

where does the less than symbol disappear too

 

[3.1415926535]

$$|x+y|\leq |x|+|y|\Leftrightarrow(|x+y|)^2\leq (|x|+|y|)^2\Leftrightarrow(x+y)^2\leq (|x|+|y|)^2\Leftrightarrow x^2+2xy+y^2\leq|x|^2+2|x||y|+|y|^2\Leftrightarrow x^2+2xy+y^2\leq x^2+2|x||y|+y^2\Leftrightarrow 2xy\leq2|x||y|$$
$$xy\leq|x||y|$$

That is obvious since

$$x\leq|x|$$ and $$y\leq|y|$$