Hey everyone.

I'm having a little trouble understanding why it is that proof by induction constitutes as proof. If anyone could spare a little of their time and discuss this with me, I'd be much obliged.

Here's my understanding...

I was given this Theorem from my abstract algebra text (I know, i'm studying abstract and I can't even prove using induction...I'm screwed). The concepts of the algebra are easy to master, but I'm american, so I've never been instruced in the art of proof.

Theorem:

Let $\displaystyle S$ be a set of positive integers with the properties

a) $\displaystyle 1\in{S}$

b) whenever the integer $\displaystyle k\in{S}$, then $\displaystyle k+1\in{S}$ must also be in $\displaystyle S$.

Then $\displaystyle S$ is the set of all positive integers.

O.K. So, the book goes on to prove this theorem by employing the Well-Ordering Principle. I understand the proof, but I'm a little fuzzy on why this theorem allowa us to verify that certain relations are true for all $\displaystyle n\in{S}$.

Let's talk guys.