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## Homework Statement

Show that the Archimedean axiom O5 follows from the Least Upper Bound Property O6, together with the other axioms for the reals.

## Homework Equations

O5 = [if a,b > 0, then there is a positive integer n such that b<a+a+a+...+a (n summands)] or [if a,b > 0, then b < na or b/a < n]

O6 = if A is any nonempty subset of

**R**that is bounded above, then there is a least upper bound for A.

## The Attempt at a Solution

My teacher told us to do this as a proof by contradiction so that's the format I'll be doing.

Suppose the Archimedean axiom is false towards a proof by contradiction. Therefore, there exists some a,b > 0 such that b [itex]\geq[/itex] na, or b/a [itex]\geq[/itex] n.

Then the set, say N, is bounded above by b/a and so sup(N) exists. Write sup(N) = S.

And then I can't figure out how to finish this proof.

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