Least upper bound
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Given a subset 
in some larger 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.
The Least Upper Bound Axiom: This is one of the fundamental axioms of real analysis. According to it, any nonempty set of real numbers that is bounded above has a supremum. This is something intuitively clear, but impossible to prove using only the field properties, order properties and completeness property of the set of real numbers.
