\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \)
deepdream of
          a sidewalk
Show Answer
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)
Math and science::Analysis::Tao::05. The real numbers

Supremum

Let \( E \) be a subset of \( \mathbb{R} \). If \( E \) is not empty and has some upper bound, we define \( sup(E) \) to be [...].

We introduce two new symbols, \( +\infty, -\infty \), to deal with two special cases. If \( E \) is non-empty and [...], we set \( sup(E) := +\infty \); if \( E \) [...], we set \( sup(E) := -\infty \).

We refer to \( sup(E) \) as the supremum of E, and denote it as sup E.