If x is a point such that every neighborhood of x contains infinitely points of S then x is a cluster point of S and x is in S’.

If S is a subset of T and every neighborhood of x contains infinitely points of S, then does every neighborhood of x contains infinitely points of T? Does that prove that S’ is a subset of T’?

We know that does that mean ? Now you can finish the second one.

All of the others have similar proofs.