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Section 21.3 The completeness property
Objectives
Compute the supremum and infimum of subsets of real numbers.
Understand completeness as the property that every bounded set has a least upper bound.
Use completeness to establish the Archimedean property and density results.
A set
\(S\) is bounded above if there is some
\(b\in\bb R\) such that
\(s<b\) for all
\(s\in S\text{.}\)
Simiilarly,
\(S\) is bounded below if there is some
\(b\in\bb R\) such that
\(b<s\) for all
\(s\in S\text{.}\)
The supremum of a set
\(S\text{,}\) denoted
\(\sup S\text{,}\) is the least upper bound of that set. That is,
\(\sup S\) is an upper bound of
\(S\text{,}\) and if
\(b\) is any upper bound of
\(S\text{,}\) then
\(b\ge\sup S\text{.}\)
The infimum of
\(S\text{,}\) denoted
\(\inf S\text{,}\) is the greatest lower bound.
A set
\(X\) has the supremum property if any subset
\(S\subseteq X\) which is bounded above has a least upper bound in
\(S\text{.}\) Such a set
\(S\) is called complete.
The real numbers are complete. The rationals are not. (Neither are the hyperreals.)
We can say that
\(\bb R\) is the completion of
\(\bb Q\text{,}\) we we have added just enough numbers to make it complete.
If
\(b=\sup S\text{,}\) then for any
\(b-\eps\text{,}\) there is somet
\(s\in S\) such that
\(s>b\text{.}\)
An ordered field is Archimedean if for any
\(a,b\) with
\(a>0\text{,}\) there exists some
\(n\in\bb N\) such that
\(nx>y\text{.}\)
In essence this says that there is no pair
\((a,b)\) such that
\(a\) is infinitesimal with respect to
\(b\text{,}\) or
\(b\) is infinitesimal with respect to
\(a\text{.}\)
The real numbers are Archimedean. The hyperreals are not.
Can be used to show that
\(\bb Q\) is dense in
\(\bb R\text{,}\) that is, for any
\(x,y\in\bb R\text{,}\) there exists
\(q\in\bb Q\) such that
\(x<q<y\text{.}\)
