[Math] Questions about completeness axiom

Question

I'm now going through the completeness axiom. What is it used for? What can you do with it? Why is it called completeness?
And beside that, how can you proof this theorem?
Suppose that S is a nonempty subset of R and k is an upper bound of S. Then k is the least upper bound of S if and only if for each $\epsilon > 0$ there exists $s \in S$ such that $k – \epsilon < s$.
I tried picking a random $\epsilon$, but then I come to the point $k-s<-\epsilon$..

Answer 1

The completeness axiom is probably the most important concept in real analysis. Every theorem in real analysis follows from it; for instance, every convergent sequence of real numbers has a real limit (which is not the case for, say, rational numbers, which are not a complete field). The fact that real numbers are a continuum (which is implied by completeness) allows you to derive most results in calculus, etc.

Also, as its name implies, the completeness axiom is an axiom, not a theorem, therefore, there's no proof for it (at least if you're using an axiomatic, that is, non-constructive, definition of real numbers).