Is the product of two random variables finite if the expectation of the random variables squared is finite?

Question

If $E X_t^2 < \infty$ and $E Y_t^2 < \infty$, then is $E X_t Y_t$ finite?

I am thinking yes, because of Cauchy-Schwartz, since $E X_t Y_t - E X_t E Y_t$ is an inner product, so it's less than something like $\sqrt{Var X_t } \sqrt{Var Y_t}$ which is finite?

But I am not sure.

Answer 1

Yes, Cauchy-Schwarz says $|\mathbb E[XY]| \le \sqrt{\mathbb E[X^2] \mathbb E[Y^2]}$.