If $X$ is a random variable satisfying $P[|X| \lt \infty]=1$, then show that for any $\epsilon >0$, there exists a bounded random variable $Y$ such that $P[X \neq Y] \lt \epsilon$.
Note: A random variable $Y$ is bounded if for all $\omega$, $|Y(\omega)| \le K$ for some $K$ independent of $\omega$.
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