1. ## Closed set

I am having trouble solving this problem from Principles of Mathematical Analysis by Rudin.

*Let $E'$ be the set of all limit points of the set $E$. Prove that $E'$ is closed.

What I am trying to do is prove that the complement of the set is open thus making the set itself closed. So far I have

$A=(E')^c$. if $x\in A$ then $x$ is not a limit point of $E$ so there exist a neighborhood $N_r(x)$ such that $N_r(x)\subset E^c$ for some $r>0$. Therefore, $\forall x\in A$ we have that $x$ is a limit point of $E^c$.

Any help or pointers on what I should do next would be greatly appreciated.

2. There is a similar question here.

3. Originally Posted by putnam120
I am having trouble solving this problem from Principles of Mathematical Analysis by Rudin.

*Let $E'$ be the set of all limit points of the set $E$. Prove that $E'$ is closed.

What I am trying to do is prove that the complement of the set is open thus making the set itself closed. So far I have

$A=(E')^c$. if $x\in A$ then $x$ is not a limit point of $E$ so there exist a neighborhood $N_r(x)$ such that $N_r(x)\subset E^c$ for some $r>0$. Therefore, $\forall x\in A$ we have that $x$ is a limit point of $E^c$.

Any help or pointers on what I should do next would be greatly appreciated.
You have gone slightly to far. What you have shown is that for all $x \in A$ there exists a neighbourhood of $x$ which contains no points of $E'$, so is a subset of $A$, which proves that $A$ is open.

RonL