The problem is as following: $A\subset X$. Show that IF $C$ is closed set of $X$ and $C$ contains $A$, then $C$ contains the closure of $A$
Here is my proof, but I dont know whether I have the right idea or not. A comment about its truth would be appreciated. It is not a homework problem, I am doing them for the hell of it.
Since $C$ is closed $\implies C$ contains its limit points. Since $A\subset C$, then $C$ contains the limit points of $A$ too. The definition of $\bar A$ is that $\bar A$ is the set $A$ and its limit points. Then, $C$ contains $\bar A$ too.
Is this proof true?
Thanks in advance!