From Williams' Probability w/ Martingales:
How exactly does that inequality hold true?
I get that $E[X_i^2] \le E[X_i^2]^2 \le K$ and $E[X_j^2] \le E[X_j^2]^2 \le K$, but how does that mean $E[X_i^2] E[X_j^2] \le K$?
I'm guessing it follows from saying $E[X_i^2 X_j^2] \le K$. If so, why is that true? Might be related to $E[X_i^4] \le K$, I guess, but I'm not quite sure how to make the inference.
We have $[\mathbb E(X_i^2)]^2\leq K$ or equivalently $\mathbb E(X_i^2)\leq K^{\frac12}$ for each $i$.
Then: $$\mathbb E(X_i^2)\mathbb E(X_j^2)\leq K^{\frac12}K^{\frac12}=K$$