Prove that if X and Y have finite second moments (i.e. E(X^2) and E(Y^2) are finite), then X+Y has a finite second moment. (X+Y)^2 ≤ X^2 + Y^2 + 2|XY| => E[(X+Y)^2] ≤ E(X^2) + E(Y^2) + 2E(|XY|) I don't understand the (probably incomplete) proof. On the right side, E(X^2) and E(Y^2) are finite, but how can we know whether E(|XY|) is finite or not? Thanks for explaining!