$E(xy)<infty$ proof

I am reviewing the best linear projection properties proof in Hansen’s book on econometrics. Specifically, the proof according to which $$E(xy). For this, it is assumed that $$E(y^2) and $$E||x||^2. The proof is as follows:
$$|E(xy)|
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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