# Is the Archimedan property of $\Bbb{R}$ the most important property in real analysis?

I am learning about sequences and for example:

A series $\sum_{k = m}^ns_k$ is convergent iff for every $\epsilon > 0[\exists N : m, n > N \Longrightarrow |\sum_{k = m}^ns_k| < \epsilon]$.

Now in this example, if it wasn't for the Archimedan property then this wouldn't hold true - there would be an infinitesimal/infinite element such that any sum is greater than said infinite(simal) element.

In fact, if it wasn't for the property, most of real analysis would fail - the definition of $\epsilon$ involves $\exists n : \frac{1}{n} < \epsilon$.

Does this also have to do with the density of $\mathbb{Q}$ in $\Bbb{R}$? Topologically perhaps?

• I think that the fact that the real numbers are a field is important. The fact that they can be given an order that is compatible with the field is important. The fact that every set of real numbers that has an upper bound has a least upper bound is important. In fact these three properties essentially determines the real numbers. – Baby Dragon Nov 24 '13 at 9:18
• It is hard to say which is "the most important property", but if I had to pick one, it would be the least upper bound axiom. – Prahlad Vaidyanathan Nov 24 '13 at 9:23
• Ah, and this is a consequence of said axiom @PrahladVaidyanathan – Don Larynx Nov 24 '13 at 9:24
• An ordered field with the least upper bound property is automatically Archimedean. In fact, $\Bbb R$ is the unique ordered field with the least upper bound property, though this is a bit harder to prove. See this answer and the comments that follow it. – Brian M. Scott Nov 24 '13 at 21:10

I would say "no". $\mathbb Q$ is Archimedean, but would be terrible for analysis. In contrast, things like the Levi-Civita field are non-Archimedean, but their non-Archimedean analysis is still studied.
What does have to do with the density of $\mathbb Q$ in $\mathbb R$ is the fact that real epsilons are unnecessary: As you point out, for all positive $\varepsilon$, you can find a positive rational number smaller than it. That flavor of statement doesn't really require the Archimedean property, though, as it's also true for instances of the Hyperreals.