Proof of converge in probability

Question

For $\epsilon>0$ define $E_{n}=\begin{Bmatrix}\omega:|X_{n}(\omega)-X(\omega)|>\epsilon\end{Bmatrix}$, If $X_{n}\overset{a.e}{\rightarrow}X$, then show that $\mathbb{P}(\cup_{n\geq m}E_{n})\rightarrow0$ where $X_{n}$ is a sequence of random variable. I know that because of "almost everywhere" convergence, some parts do not really converge, and these parts have measure of 0, but how can I prove the statement above?

Answer 1

Let $A_n=\cup_{n \geq m} E_n$ and $A=\cap_{n \geq 1}A_n$. Verify that $A$ is contained in $\{\omega: X_n(\omega) \text {does not tend to} X(\omega)\}$. By hypothesis we have $P(A)=0$. Now verify that $\{A_n\}$ decreases to $A$. It follows that $P(A_n) \to 0$.