Structure theorem for closed set in R

Question

Any Closed set in $ \mathbb R$ can be given by intersection of countable open set.

My attempt:

Let $E$ be closed set in $\mathbb R$.

• Case 1 : $E=\emptyset$.

$$E=\bigcap_{n\in \mathbb N} \left(0,\frac{1}{n}\right)$$

Case 2: $E\neq \emptyset $

$$G_n=\bigcup_{x\in E} B\left(x,\frac{1}{n}\right)$$

As $G_n$ is union of open ball hence $G_n$ is an open set.

Claim:

$$E=\bigcap_{n\in \mathbb N} G_n $$

$E\subset \bigcap_{n\in \mathbb N} G_n $ directly from definition.
Let $t\in \cap_{n\in \mathbb N} G_n $ so $t\in B(x,1/n)$ for some $x \in E $ and $\forall n\in \mathbb N$. Hence $t$ is limit point of $E$ and hence $t\in E$ as $E$ is closed.
$\bigcap_{n\in \mathbb N} G_n \subset E$.
$\bigcap_{n\in \mathbb N} G_n =E$

Answer 1

The proof is correct (apart from a few errors of English grammar). However, the phrase "so it can be given by countable open intertval by Lindel theorem" is not needed.