Let $X$ and $Y$ be (possibly dependent) random variables. I want to show that if the variance of $X$ and $Y$ both go to zero, the variance of the product $XY$ also goes to zero. In other words, for any small $\epsilon$, we can make $Var(XY) < \epsilon$ by making $Var(X)$ and $Var(Y)$ small enough.
I have tried to show this by expanding into expectations but I can’t seem to get the result I want.
It's not true. Try $X=Y= \delta^{-1/3}$ with probability $\delta$, $0$ otherwise.