If $X$ and $Y$ are independent continuous random variables with densities $f(x)$ and $g(y)$ then the probability that $Y$ is less than or equal to $X$ is
$\Pr (Y \le X) = \int _{x=-\infty}^{\infty}(f(x)\int _{y=-\infty}^{x}g(y)dy)dx$
Say $f(x)$ and $g(y)$ are parameterized with $n$ and we have $Var(X)$ and $Var(Y)$ converging to zero as $n\to\infty$, where at all times $E[Y]-E[X]>\epsilon$ for some $\epsilon>0$. How do we prove that, $\Pr (Y \le X)\to0 $ as $n\to\infty?$