# Var(XY) goes to zero if Var(X) and Var(Y) go to zero

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.
• You can adjust these slightly so the expected values are $0$: take $X=Y=-\delta^{2/3}/(1-\delta)$ otherwise, instead of $0$. – Robert Israel Nov 21 '16 at 7:18