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If $X$ is uniformly bounded and $\lim_{n \to \infty} X_n=X$, a.s then $X_n$ is uniformly bounded?

For real sequences we know that every convergent sequence is bounded.So for every $\omega$ we can say that $\vert X_n(\omega) \vert <M(\omega)$. Buy how can I conclude the existence of a uniform constant $M'$ for all $\omega \in \Omega$. such that $\vert X_n\vert <M'$.

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This is false. On the space $(0,1)$ with Lebesgue measure let $X=0$ and $X_n=nI_{(0,1/n)}$. Then $X_n \to X$ at every point, $X$ is bounded but $\{X_n\}$ is not uniformly bounded.

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