But your proof is not clear. Don't try to make it shorter. x is a limit point of $\displaystyle E^c$ but $\displaystyle x \not \in E^c$ ? Why ? Shouldn't you say "Let's assume $\displaystyle x \not \in E^c$" ?$\displaystyle \color{red}\text{I did, I said lets assume its a limit point of }E^c\text{ but not in }E^c$

And I think at least that red part should be changed. It is no sense to say that a set "belongs" to a set (in that case). $\displaystyle \color{red}\text{What you have put in red I did not have in my post Moo}$

Also, I've never been fond of «"blabla" because there is a contradiction if». It does not follow a logical path. Make your assumptions, state clearly what you suppose. List the properties that would come from these assumptions and then

**conclude** by saying there is a contradiction.

By definition I proved that a limit point of E^c could not be an element of E because it would contradict E being open (since every point of an open set must be an interior point), So since all limit points of E^c can not be in E they must be in E^c because they are compliments, and the defintion of a closed set is a set who contains all its limit points. And I wrote this post with definitions and references specifically for the OP. I wrote it for them, not other people. So sorry if I made references to things that others are not well-versed in. I did not want to write a formal proof and do the homework for them, I wanted to give an outline. Later I will come back and write a formal proof with definitions. Thank you Moo, I appreciate when people call me out on things (even though here I think I was ok) because its forces me to get better. P.S. you misquoted me somehow.