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 , otherwise, nothing needs to be proved)" (The propositions referenced basically establish the linearity of dot-product, and that the norm is always non-negative.)

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

but I don't see why I'm supposed to believe this.

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