is what you are trying to prove by induction, NOT something you already know to be true! You want to prove . How does it go from there?
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:
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.