Proof by mathematical induction.

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.

Re: Proof by mathematical induction.

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?

Re: Proof by mathematical induction.

Quote:

Originally Posted by

**godelproof**

That is where I seem to mess up. I haven't dealt with inequalities in my problem sets.

So, I'll assume that:

and . Showing that is not true would prove validity of the original statement, correct?

Re: Proof by mathematical induction.

What I don't understand his how you explain the statement . I mean from where does the power change from 2 to 3. Also I think that the statement should be written as not .

Re: Proof by mathematical induction.

Quote:

Originally Posted by

**BAdhi** What I don't understand his how you explain the statement

. I mean from where does the power change from 2 to 3.

That is how it asks in my book.

Quote:

Originally Posted by

**BAdhi** Also I think that the statement should be written as

not

.

Yes, you're correct with the notation. I made a typo.