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Here is the question: Suppose $X$ is a $\Bbb R$-valued random variable. Show that for all $\varepsilon > 0$, there exists a bounded random variable $Y$ such that $P (X \neq Y ) <\varepsilon$.

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Try $Y=X\,\mathbf 1_{|X|\leqslant x}$ for $x$ large enough. Then $\mathbb P(Y\ne X)=\mathbb P(|X|\gt x)$. Furthermore, $\mathbb P(|X|\gt x)\to0$ when $x\to+\infty$, hence for $x$ large enough, $Y$ solves the question.

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    @Did Yeppp, now understood.2017-08-15