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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.

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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
up vote 4 down vote accepted

One way I have seen the Archimedean Property posed, which makes it relatively simple to understand, is as these two equivalent properties:

(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 $\frac{1}{n} < \epsilon$.

Perhaps you could phrase your explanation to him in terms of using the natural numbers to control how large/small the elements of the real numbers can get? Maybe an example of what would happen if the property did not hold would be best, since it demonstrates the property's usefulness?

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