Well-Ordering Principle on Natural Numbers

[Klungo]

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

Homework Statement

If [itex] n \in N[/itex], then [itex]n ≥ 0[/itex].

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

2. Relevant

(i) [itex]0 \subset N[/itex]

(ii) [itex]n+1 \in N[/itex] for all [itex] n \in N[/itex]

(iii) [itex]n-1 \in N [/itex]for all [itex] n \in N [/itex]such that n≠0

(iv) The well-ordering principle itself.

The Attempt at a Solution

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

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

I used (iii) [itex] n-1 \in N [/itex] for all n in N such that n≠0. But

[itex](n=0) \in N[/itex] and [itex]-1 \notin N[/itex]. Hence, [itex] -1<n → 0≤n[/itex]

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?

 

[mighty2000]

Thank you for sharing your proof and post. It's great to see that you are using the well-ordering principle to prove this statement.

Your proof looks good so far. I just have a few suggestions for clarification.

Firstly, when you use the notation "n-1 \in N for all n \in N such that n≠0," it would be helpful to explicitly state that n is a natural number. So the statement would be "for all n \in N such that n≠0, n-1 \in N." This makes it clear that you are only considering natural numbers.

Secondly, to address your concern about -1 \notin N, you can simply state that n-1 \in N for all n \in N such that n>0. This way, you are only considering natural numbers greater than 0, and the statement holds true for all of them.

Finally, to answer your question about writing text in between {itex} text {/itex}, you can use the {itex} command to indicate that the text inside is in italics. For example, if you wanted to write "This is an example," you would write {itex}This is an example{/itex}, and it would appear as "This is an example." Alternatively, you can use the {tex} command to indicate that the text inside is in standard font. For example, if you wanted to write "This is an example," you would write {tex}This is an example{/tex}, and it would appear as "This is an example."

I hope this helps. Keep up the good work with your proofs and posts! As you continue to practice, you will definitely get better at using tex.