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 $k-s<-\epsilon$..
Questions about completeness axiom
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real-analysis
axioms
. $$\begin{align}{}\\\end{align}$$ The axiom of completeness is that every bounded subset of $\mathbb R$ has a least upper bound.– 2011-10-26