Let $X_1, X_2, ...$ be random variables with $\mbox E X^2_j = 1$ for all $j$. Show that $\inf P[|X_n| > \epsilon] > 0$ for some $\epsilon > 0$ if and only if $\inf \mbox E |X_n| > 0$.
I have shown necessity, which is quite easy and doesn't use the hypothesis that $\mbox E X_j ^ 2 = 1$. I'm stuck on sufficiency, and can't seem to figure out how to work $\mbox E X_j ^ 2 = 1$ into anything useful.
Thanks.