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$E(xy)<infty$ proof

I am reviewing the best linear projection properties proof in Hansen’s book on econometrics. Specifically, the proof according to which $E(xy)<infty$. For this, it is assumed that $E(y^2)<infty$ and $E||x||^2<infty$. The proof is as follows:
$$|E(xy)|<E|xy|<sqrt(E|x|^2)sqrt(Ey^2)<infty$$
Where $ x $ is a vector of random variables (dimension $k$x$1$) and $ y $ a random variable. $|x|$ is the norm or euclidian length.
The first inequality is based on Expectation Inequeality and the first equality is based on the Cauchy-Schawarz Inequality. $$$$ Two questions; First, since $E (xy)$ is not explicitly in the inequality, from what part of it is its finiteness deduced? Second, all the members of the inequality are real numbers, however $E(xy)$ is a vector of dimension $k$x$1$, how then can this comparison be made?

Cross Validated Asked on January 7, 2022

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