I've been scratching my head over this for about 45 minutes now, and I have no idea where why the $w$ in this proof disappeared.
$\left| \left \langle u,v \right \rangle \right| \le \|u\| \|v\| $
We have an orthogonal decomposition $u = \frac{\left \langle u , v \right \rangle }{\|v\|^2 }v+ w $ where $w$ is orthogonal to $v$. The Pythagorean theorem states:
$$\|u\|^2 = \left\|\ \frac{\left \langle u , v \right \rangle }{\|v\|^2 }v \right\| ^2 + \|\ w \|\ ^2 $$
Next we take the square root on and multiply by $\|v\|^2$ and we are left with the inequality we wanted to prove. Where did the $w$ go?!