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Let $X_n$ be a sequence of random variables on a probability space $(Omega, F, P)$. Let’s define:

$$A={ w in Omega: lim_{n to infty} X_n(w) space text{exists}} .$$

Now, I need to show that A is an event, in other words that A is in the $sigma$-algebra F. What would be the classical approach to do that?

## Answer

Sequence $(X_n(omega))_n$ has a limit iff it is a Cauchy sequence, so:

$$A=bigcap_{n=1}^{infty}bigcup_{k=1}^{infty}bigcap_{r=k}^{infty}bigcap_{s=k}^{infty}left{omegainOmega: left|X_{r}left(omegaright)-X_{s}left(omegaright)right|<n^{-1}right}$$

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