Question

prove that the derived set of a set is a close set

Answer 1

I am following a proof of the statement

The derived set(the set of accumulation points) A′A′ of an arbitrary subset AA of R2R2 is closed.
in a book.

It starts with

Let qq be a limit point of A′A′. If it is proved that q ∈A′∈A′, then the proof is done.

Let GqGq be the open set containing qq . Since qq is a limit point of A′A′,GqGq contains at least one point r∈A′r∈A′ different from qq. But GqGq is an open set containing r∈A′r∈A′; (Up to this I understood) hence GqGq contains infinitely many points of AA (How? I did not get this.)

So there exist a∈Aa∈A such that a≠q,a≠ra≠q,a≠r and a∈Gqa∈Gq. That is,each open set containing qq contains infinitely many points of AA. Hence q∈A′q∈A′.

Can you help me out.