A way to teach Archimedean property

Question

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.

Answer 1

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?