Does $|X_n|\le\Delta_n+\delta$, with $\Delta_n\overset{p}\to0$ imply $|X_n|\overset{p}\to0$?

by J.Mike   Last Updated May 15, 2019 18:20 PM

Suppose $\Delta_n\overset{p}\to0$, and for any $\delta>0$, we have $$|X_n|\le\Delta_n+\delta.$$

Can we conclude that

$$|X_n|\overset{p}\to0?$$

Here $\Delta_n\overset{p}\to0$ means that for any $\varepsilon>0$, we have $\lim_{n\to\infty}P(|\Delta_n|>\varepsilon)=0$.



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