# Schwarz inequality and general proof

Recently I've learn about the Cauchy-Schwarz inequality. So, as we all know this this inequality states that for an inner product vector space for all vectors $$x, y$$ we have that:

$$|\langle x,y\rangle|\le ||x||\text{ }||y||$$.

The proof of this inequality is well known but I have trouble with understanding something. I mean if this inequality works for any inner product vector space we just have to proof it once, yes? What I mean is that for different vector spaces this inequality takes different form but these are just special cases, right? If we proved that this inequality works in any inner product vector space then we, know that it works for example in $$\mathbb{R}^n$$ with dot product right? This is just special case of general result so we don't need a special proof, right? So all the proofs that can be found for example here Proofs of the Cauchy-Schwarz Inequality? for the lack of a better world are made for "fun"?

Do I get it right?