Validity of proof of Cauchy-Schwarz inequality

[HaniZaheer]

Proof: If either x or y is zero, then the inequality |x · y| ≤ | x | | y | is trivially correct because both sides are zero.
If neither x nor y is zero, then by x · y = | x | | y | cos θ,
|x · y|=| x | | y | cos θ | ≤ | x | | y |
since -1 ≤ cos θ ≤ 1

How valid is this a proof of the Cauchy-Schwarz inequality?

 

[FactChecker]

It looks good as long as x · y = | x | | y | cos θ is given by definition or is already proven. (It's good form to indicate "by definition" or "by Lemma xxx", etc.)

 

[HaniZaheer]

Alright, thanks a lot