conclude that the closure of a union is the union of the closures.

For this question, we are allowed to assume the following is true and has been proven:

Let A,B$\displaystyle \subseteq$R and p$\displaystyle \in$R. Prove that p is an accumulation point of A $\displaystyle \cup$B if and only if p is an accumulation point of A or p is an accumulation point of B: (A$\displaystyle \cup$B)'=A' $\displaystyle \cup$ B'.

Now since we assumed the above is true,

Conclude that the closure of a union is the union of the closures:

(A $\displaystyle \cup$ B)^cl=A^cl $\displaystyle \cup$ B^cl. How would I go about proving this?