Topology of derived sets and subset

Question

if A is a subset of C ,then A' is a subset of C' A' denotes set of all limits points of A

Answer 1

This is simply a matter of definitions.

$A \subset C$ means for every $x \in A$ then it is also true that $x\in C$.

And every $x \in A'$ is a limit point of $A$.

A limit point means that every neighborhood of $x$ contains a point $y; y\ne x$ so that $y \in A$.

But $A \subset C$ so $y \in C$.

So every neighbor hood of any $x \in A'$ contains a point $y; y \ne x$ so that $y \in C$.

So any $x\in A'$ is a limit point of $C$.

So for any $x \in A'$ it is also true that $x\in C'$ and $A' \subset C'$.

There was nothing actually to prove. It was just a matter of stating and understand what definitions mean.