# A way to teach Archimedean property

A student asked me how to understand the Archimedean property, I tried to re-read with him what he has already done in class (well, actually copy from the blackboard in class). However I think I'm not helping much, my suspicion is that maybe I'm being too formal. How can I approach this differently? He just started studying college, and I think he's not very familiarized with math language, I belive that this is the first "formal" prove they have encounter in class, the rest has been very empirical and easy going.

-
What level of student do you teach? High school? What is this student's background? – Potato Aug 29 '13 at 20:28
@Potato I was about to make an edit with that information. – Ana Galois Aug 29 '13 at 20:30
Thanks. For the purposes of this question, it is very important information! – Potato Aug 29 '13 at 20:33
Informally, I interpret the Archimedean property to mean: If you give me any real number, then I can think of a bigger natural number. – Adriano Aug 29 '13 at 20:40
I think non-examples are very useful in the understanding of a definition or condition. Introduce the student to a case he/she can understand where the order doesn't follow the Archimedean property. Then go over the order on the real line and see why the proof you've already showed him/her works. – Jonathan Y. Aug 29 '13 at 20:40

(1) For any positive number $c$, there is a natural number $n$ such that $n >c$.
(2) For any positive number $\epsilon$, there is a natural number $n$ such that $\frac{1}{n} < \epsilon$.