How to understand that both the lim inf and lim sup of the sets of real numbers is either open or closed?

I am reading the Real Analyais (4th edition) by Royden, H. L., & Fitzpatrick, P. In Section 1.4, it is said that

both the lim inf and lim sup of a countable collection of sets of real numbers, each of which is either open or closed, are a Borel set.

I do not know how to obtain this result. The lim inf and lim sup of sets of real numbers are defined as follows.

Let $\{A_n\}_{n=1}^{\infty}$ be a countable collection of sets that belong to a $\sigma$-algebra $\mathcal{A}$. Define the sets\begin{align*} \limsup \{A_n\}_{n=1}^{\infty} &= \bigcap_{k=1}^{\infty} \left( \bigcup_{n=k}^{\infty} A_n \right) , \\ \liminf \{A_n\}_{n=1}^{\infty} &= \bigcup_{k=1}^{\infty} \left( \bigcap_{n=k}^{\infty} A_n \right) .\end{align*}