Best approximation proof question

Question

Here is proof on best approximation , V is a finite-dimensional inner product space, $W$ is a subspace of $V$, $x \in V$ $\|{x-P_wx}\| \leq\|x-y\|$ for $y \in W$ , where $P_wx$ is the orthogonal projection linear transformation from $V$ onto $W$

$\|x-y\|^2=\|x-P_wx+P_wx-y\|^2$
$=\|x-P_wx\|^2+\|P_wx-y\|^2$ since $x-P_wx$ and $P_wx-y$ $ \in W$are orthogonal
$\geq \|x-P_wx\|^2$

My question is, for the last line, how did the $\|P_wx-y\|^2$ term disappear? I know that it is positive but how does it mean $\|{x-P_wx}\| \leq\|x-y\|$?

Answer 1

It disappeared simply because it is nonnegative.

$$x+y \ge x$$

holds whenever $x,y \ge 0$ for that reason. You can see this by working backwards and adding $x$ to both sides:

$$y \ge 0 \implies x+y \ge x$$

In this case, you're doing that with norms, which are nonnegative. From that step in the proof you mentioned, then, taking the square root of each side gives the desired result.