Definition
26
[20H] (Solved on 2022-11-24) Given \(J\) an index set (not empty), let \(a_ nββ\) for \(nβ J\). The supremum and infimum are defined as
\[ \sup _{nβ J}a_ n = \sup A \quad ,\quad \inf _{nβ J}a_ n = \inf A \]
where \(A=\{ a_ n: nβ J\} \) is the image of the sequence.
Given \(D\) not empty, let \(f:Dββ\) be a function. The supremum and infimum are defined as
\[ \sup _{xβ D}f(x) = \sup A \quad ,\quad \inf _{xβ D}f(x) = \inf A \]
where \(A=\{ f(x): xβ D\} \) is the image of the function.