You prove A > B by showing A − B is something you already know can't be negative. Drag the slider to see the gap between two vectors and the Cauchy-Schwarz bound that guarantees it stays positive.
The universal starting move: rearrange until you're proving something ≥ 0. Expand (a − b)² and you get a² − 2ab + b² which is never negative because it's literally a square. This lets you prove a² + b² ≥ 2ab instantly. Stack this trick into higher dimensions and you get Cauchy-Schwarz: the square of the dot product can never exceed the product of the squared magnitudes. Every classical inequality is this move wearing a different disguise.
Proving an inequality means constructing a chain of steps that ends in something obviously non-negative, usually a perfect square or a norm.