Use induction in a non standard way

[Dmobb Jr.]

So I have to do an induction but I am not quite sure how to set it up. I have already proven that at each step I have either my intended result or I can advance one more step. I have also proven that there are a finite number of steps.


Intuitively I have essentially completed the proof. I just can't figure out how to present this in a way that is completely rigorous.

I get the feeling that "Well there can only be finitely many steps so eventually it will happen" is not good enough.

 

[HallsofIvy]

I don't see how anyone can suggest anything when what you are trying to do is so vague.

 

[Dmobb Jr.]

Well I feel a little bit guilty for even posting a homework question at all (It's cheeting). So I am not going to post specifics. I will clarify certain things if people have questions. I think that someone could figure this out with the information I have given. If not then I will just have to figure it out myself which is what I should be doing anyway.

 

[rideabike]

Sounds like a good "dot-dot-dot" proof, but maybe your professor doesn't like those. I.e. show the process of a couple steps, then "dot-dot-dot", then show the final step!

 

[verty]

No, I was going to ask questions but this is so vague, it isn't in the ballpark of meaning anything.

 

[Dmobb Jr.]

Yeah i wish I could dot dot dot this one but that's definitely not allowed. I reallized that I had messed up at an earlier part of this problem anyway. Thanks though.

@verty While what I said was not 100% rigourus I could easily make it that way.

For all natural numbers n, if x is not $\geq$ n, then x < n. Also there exists Y$\in$N such that x $\leq$ Y.

Prove that x exists and is a natural number.

Edit: We must assume x is a natural number not prove it.