Cauchy-Schwarz Inequality Proof

So I'm reading this document that has a proof of the Cauchy-Schwarz Inequality, and I'm missing the logic.

It says, "By Proposition 2.3 and (i) of Proposition 2.5, we have (assuming $\displaystyle y \ne 0$, otherwise, nothing needs to be proved)" (The propositions referenced basically establish the linearity of dot-product, and that the norm is always non-negative.)

$\displaystyle 0 \leq ||x - \lambda y||^{2} = ||x||^{2} - 2 \lambda x \cdot y + \lambda^{2} ||y||^{2} \leq (||x|| - \lambda ||y||)^{2} + 2\lambda\big( ||x|| \, ||y|| - |x\cdot y|\big)$.

I get everything here except the last inequality. I see that it reduces to this:

$\displaystyle -2 \lambda x \cdot y \leq -2 \lambda |x \cdot y|$ but I don't see why I'm supposed to believe this.

Also, tangentially, can anyone tell me what's wrong with this proof?:

$\displaystyle ||x||\, ||y|| = (x \cdot x)^{1/2} (y \cdot y)^{1/2} = (x \cdot x \cdot y \cdot y)^{1/2} = (x \cdot y)^{1/2}(x \cdot y)^{1/2} = |x \cdot y|$