Show that for $n∈N$ and $\epsilon>0$ there exists $R>0$ such that $P(|X_n|>R)<\epsilon$. Suppose $X_n$ is a sequence of random variable, taking value from ($-\infty,\infty$). Show that for $n∈N$ and $\epsilon>0$, there exists $R>0$  such that $P(|X_n|>R)<\epsilon$.
I mean, intuitively it is true since the value of $X_n$ is finite. We can just let $R$ be large enough and then $P(|X_n|>R)<\epsilon$.
But I do not know how to officially prove it. Anyone could help me on it?

We want to show for $R$ large enough, $$\epsilon>P(|X_n|>R)=1-P(|X_n|\leq R)=1-F_{|X_n|}(R)\iff F_{|X_n|}(R)>1-\epsilon,$$
where $F_{|X_n|}$ is the CDF of $|X_n|.$ By properties of CDFs, the limit of a CDF is $1$ as its argument tends to $+\infty$, so the desired result follows immediately.
