convergence in distribution and convergence of sequence of real number

Question

Suppose that $X_n \rightarrow X$ in distribution and let $F_n$ and $F$ denote the cdf's of $X_n$ and $X$ respectively. I want to show that, for any sequence of real numbers $(a_n)$, if $a_n \rightarrow \infty$ then $F_n(a_n) \rightarrow 1$ and similiarly, $F_n(a_n) \rightarrow 0$ when $a_n \rightarrow -\infty$.

It seems obvious but I don't know how to show that. I know that if $a_n \rightarrow a$ and $a$ is a continuity point of $F$, then $F_n(a_n) \rightarrow F(a)$. Any help would be appreciated.

Answer 1

Let $\epsilon >0$ and choose a continuity point $a$ of $F$ such that $F(a)>1-\epsilon$. Choose $m$ such that $F_n(a)>F(a)-\epsilon$ for $n \geq m$. Choose $k$ such that $a_n >a$ for $n \geq k$. Then, $n \geq \max \{m,k\}$ implies $F_n(a_n) \geq F_n(a)>F(a)-\epsilon>1-2\epsilon$.

The second part is similar.