Limits and Continuity, Least Upper Bound
Least Upper Bound
Let S be a nonempty set of real numbers.
The supremum and infimum of S are the least upper bound
and greatest lower bound respectively.
If S is bounded above there is a unique supremum,
and if S is bounded below there is a unique infimum.
This is usually stated as an axiom,
the beginning of real analysis,
but if real numbers are defined properly,
this principle can be
proved rigorously.