Proof of Cauchy-Schwarz Inequality in probability form

Question

In my university course, we were given the following proof of the Cauchy-Schwarz Inequality:

My issue is with the last line, surely we get that:

$$|E(XY)| \leq \sqrt{E(X^2)E(Y^2)}$$ but it is not true in general that $E(|XY|)\leq |E(XY)|$

Any help would be much appreciated.

Answer 1

The result follows from the fact that X and Y are assumed to be non-negative, as stated in the first line.

Answer 2

Simply apply what you have proved with $X,Y$ replaced by $|X|,|Y|$.