Existence of Expectation of Product of Two Random Variables In the theorem's proof (See screenshot link; StackExchange does not yet allow me to directly insert images), the author uses the linearity of expectation. However, linearity of expectation requires that each individual expectation being summed is finite. How do we know that E(XY) is finite given only that both variances are finite? Thank you!
Proof


Cauchy-Schwarz inequality:
$$
E\left[|XY|\right]\le\sqrt{E[X^2]\cdot E[Y^2]}.
$$
