proof of existence of z in x < z < y
If x and y are arbitrary real numbers with x < y, prove that there is at least one real z satisfying x < z < y.
At my disposal, the least-upper-bound axiom - every nonempty set S of reals bounded above has a supremum. And a handful of theorems including x > sup S - h where h > 0 and set S has a supremum.
Is my proof OK?
Let the set S be bounded above by y, so y = sup S. Let x S and x + h = y = sup S. Then by the above theorem, there exists a z > sup S - h, therefore sup S = y > z > sup S - h = x. Q.E.D.