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Consider a sequence of real random variables $(X_n)$ defined on a common probability space and let $S\subset \Omega$ be such that $P(S)=1$.

For any $\omega \in S$, assume that every subsequence of $(X_n)$ contains a further subsequence $(X_{n_k})$ such that $\lim_{k\to\infty} X_{n_k}(\omega) = x(\omega)$.

Can I conclude that $X_n\to x$ almost surely?


My understanding of the above property on the subsequences is that $\lim_{n\to\infty} X_{n}(\omega) = x(\omega)$ for all $\omega \in S$. But if this is true, then we have obtained just the definition of almost sure convergence of the sequence $(X_n)$ to $x$ because $S$ has measure one. Is this correct?

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