[Math] If $x$ is the limit point of a finite union then is $x$ a limit point of some set in the union

real-analysis

Background

This question arose from the following problem:
If $B_n=\bigcup_{i=1}^{n}A_i$ then prove that $\overline{B_n}=\bigcup_{i=1}^n \overline{A}_i$ for $n=1,2,3…$. By $\overline{X}$ we mean the closure of $X$, by $X'$ we mean the set of limit points of $X$ and by $N_r(x)$ we mean the neighborhood of $x$ of radius $r$.

Question and Attempt

I have done this problem before by induction but today I wanted to try doing it directly.
Here is what I did:

We have to show that $$\left( \bigcup_{i=1}^n A_i \right)\bigcup\left( \bigcup_{i=1}^n A_i \right)'= \bigcup_{i=1}^n \left(A_i \cup A'_i \right) .$$ If $x \in \left( \bigcup_{i=1}^n A_i \cup A'_i \right)$ Then it is not too hard to show $x \in \left( \bigcup_{i=1}^n A_i \right)\bigcup\left( \bigcup_{i=1}^n A_i \right)'$. On the other hand though, suppose $x \in \left( \bigcup_{i=1}^n A_i \right)\bigcup\left( \bigcup_{i=1}^n A_i \right)'$ and consider the case when $x \in \left( \bigcup_{i=1}^n A_i \right)'.$ Then for all $r$, $N_r(x) \bigcap \left(\bigcup_{i=1}^n A_i \right)$ is not empty (in fact it contains infinitely many points since $x$ is a limit point). So for any $r$, this intersection contains infinitely many points of some $A_j$ ($1 \leq j \leq n$).

How do we then conclude that $x$ is a limit point of some $A_j$, bearing in mind that as $r$ is changed, $N_r(x)$ may intersect a different set $A_k$?

Best Answer

Yes.

Suppose not. Let $x$ be a limit point of $ A\cup B$, but not a limit point of $A$ nor $B$. Then exist $r_1,r_2>0$, s.t. $B_{r_1}(x)\setminus \{x\}\cap A =\emptyset$ and $B_{r_2}(x)\setminus \{x\}\cap B =\emptyset$. Then take $r=\min \{r_1,r_2\}>0$, we get $B_{r}(x)\setminus \{x\}\cap (A\cup B) =\emptyset$. Contradiction.

This does not generalize to infinite union. For example, $0$ is a limit point of $\bigcup [\frac{1}{n},1]$, but it's not a limit point for any of them.

You can also prove the theorem you want by using closure of $E$ is the smallest closed set containing $E$. Hence $\overline{B_n} \subset \bigcup_{i=1}^n \overline{A}_i$ since finite union of closed set is closed. You can easily get the other side inclusion.

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