# Well-Ordering Principle on Natural Numbers

1. Jan 13, 2012

### Klungo

This is my first proof and post. I'll eventually get better at tex.

1. The problem statement, all variables and given/known data

If $n \in N$, then $n ≥ 0$.

Hint: $N \subset N$ (thus not any empty set) and has least member by the well-ordering principle.

2. Relevant

(i) $0 \subset N$

(ii) $n+1 \in N$ for all $n \in N$

(iii) $n-1 \in N$for all $n \in N$such that n≠0

(iv) The well-ordering principle itself.

3. The attempt at a solution

As the hint suggests, I am supposed to prove this using the well-ordering principle.

$n-1 \in N$ for all $n \in N$ such that n≠0. $N \subset N$ so N≠∅ and has a least member by well-ordering principle.

I used (iii) $n-1 \in N$ for all n in N such that n≠0. But

$(n=0) \in N$ and $-1 \notin N$. Hence, $-1<n → 0≤n$

I'm not sure if this is right though.

Additionally, is there any way to write text inbetween {itex} text {/itex} such that notations appear in addition to original text?

Last edited: Jan 13, 2012