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Section Summary
Important ideas that we discussed in this section include the following.
A lower bound or a nonempty subset \(S\) of \(\R\) that is bounded below is a real number \(m\) such that \(m \leq s\) for all \(s \in S\text{.}\) A greatest lower bound (or infimum) for a nonempty subset \(S\) of \(\R\) that is bounded below is a real number \(m\) such that
\(m\) is a lower bound for \(S\) and
if \(k\) is a lower bound for \(S\text{,}\) then \(m \geq k\text{.}\)
An upper bound for a nonempty subset \(S\) of \(\R\) that is bounded above is a real number \(M\) such that \(M \geq s\) for all \(s \in S\text{.}\) A least upper bound (or supremum) for a nonempty subset \(S\) of \(\R\) that is bounded above is a real number \(M\) such that
\(M\) is an upper bound for \(S\) and
if \(k\) is an upper bound for \(s\text{,}\) then \(M \leq k\text{.}\)
The distance from a point \(x\) to a set \(A\) in a metric space \((X,d)\) is \(d(x,A) = \inf \{d(x,a) \mid a \in A\}\text{.}\) There may be no point \(a \in A\) such that \(d(x,A) = d(x,a)\text{,}\) so it is necessary to use an infimum to define this distance.
