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I saw only the proof by Lindelof Convering Theorem. But I have different arguments:

My attempt: Suppose $V = (v_1,v_2,...)$ is a collection of open sets and let's define point $p \in \cap V_i$. By definition intersection means, that $p \in V_i$ for $i = 1...n$. Also we know, that every point in $R$ is a limit point. Because $p \in V_i$ we can find $N_\epsilon(p)$ s.t $N \subset \cap V_i$ (Actually here I have proved, that intersection of finite collection of open sets are open). Because $p$ is arbitrary and is a limit point => set $\cap V_i$ contains every limit point => set is closed

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    $\begingroup$ The intersection of a finite collection of open sets is open. $\endgroup$ – Lord Shark the Unknown Jun 9 '17 at 16:52
  • $\begingroup$ How can you "define" a point in the intersection of a given set of sets? If you prove the intersection is not empty then you can take a point there, but not "define it"... $\endgroup$ – DonAntonio Jun 9 '17 at 16:54
  • $\begingroup$ finite --> countable makes it true, but this arument shows nothing. $\endgroup$ – Henno Brandsma Jun 9 '17 at 17:05
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    $\begingroup$ The phrase "is a limit point" by itself has no meaning. You have to say "a limit point of a particular set." It's true that every point in R is a limit point of R. That doesn't help you prove that it's a limit point of A. $\endgroup$ – Nate Eldredge Jun 9 '17 at 17:30
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    $\begingroup$ And now your argument is really wrong, because you when you write $V = \{V_1, \dots, V_n\}$ you are assuming that $V$ is finite. $\endgroup$ – Nate Eldredge Jun 9 '17 at 17:32
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For every subset $A$, the function $f:X \to \mathbb{R},f(x) = d(x,A) = \inf\{d(x,a): a \in A\} $ is continuous. This holds in all metric spaces.

A set $A$ is closed iff $d(x,A) = 0 \leftrightarrow x \in A$.

So for a closed $A = f^{-1}[\{0\}] = \cap_n f^{-1}[(-\frac{1}{n}, \frac{1}{n})]$ is a $G_\delta$ in any metric space.

I don't know of any proof via Lindelöfness (a metric space or its closed subsets are Lindelöf iff they are separable), that seems restrictive.

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