Show that the statement holds for all positive integers n:

Firstly, I'll start with

Therefore, 1 ϵ S.

Assuming k ϵ S:

But, k ϵ S implies that k + 1 ϵ S, therefore:

Simplify:

Therefore, k + 1 ϵ S and the statement holds for all positive integers n and all positive integers are in S.

Have I proven this correctly by using mathematical induction? I haven't dealt with an equality yet in all the problem sets that I've done before. Thanks in advance.