Bounded random variables, convergence to zero - does supremum converge to zero? Relation to Popoviciu's inequality on variances

Question

Suppose that $X_n \stackrel{p}\to 0$, and we have that $|X_n| < M$ for all $n$ and for some $M > 0$.

By Popoviciu's variance inequality, we get that, for each $n$, $$\operatorname{Var}(X_n) \leq \frac{(\sup_n X_n - \inf_n X_n)^2}{4}$$

Is it true that $\lim_\limits{n\to \infty}\sup\limits_n X_n - \inf\limits_n X_n \rightarrow 0$? This would establish that $\operatorname{Var}(X_n) \rightarrow 0$.

My worry here is that $X_n$ might take some non-zero value with probability zero, thus getting a supremum or infimum not tending to zero, despite the fact that $X_n \stackrel{p}\to 0$.

Answer 1

Let $X_n=M$ with probability $1/n$ and $0$ otherwise. Then $X_n$ converges to $0$ in probability, but $\sup_n X_n- \inf_n X_n=M$ for all $n$, so that you cannot obtain that the variance goes to $0$ using this inequality.

However, the sequence $(X_n^2)$ is uniformly integrable (because uniformly bounded), so that having $X_n\to 0$ in probability also implies that $X_n$ converges to $0$ in $L_2$. This implies that the variance goes to $0$ as well.