Least upper bound
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Given a subset 
in some larger partially ordered set 
, a least upper bound or supremum, for is an element 
such that 
for every 
and there is no 
with this same property.
If the least upper bound 
of is an element of , it is also the maximum of . If 
, then has no maximum.
Completeness: This is one of the fundamental axioms of real analysis.
A set is said to be complete if any nonempty subset of that is bounded above has a supremum.
The fact that 
is complete is something intuitively clear but impossible to prove using only the field and order properties of
