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


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

We introduce two new symbols, \( +\infty, -\infty \), to deal with two special cases. If \( E \) is non-empty and has no upper bound, we set \( sup(E) := +\infty \); if \( E \) is empty, we set \( sup(E) := -\infty \).

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

This definition simply creates syntax to refer to the least upper bound, which was shown to exist in the 'existence of least upper bound' theorem. To insure that the newly defined supremum always exists, \( +\infty \) and \( -\infty \) are introduced to handle the two edge cases where a least upper bound doesn't exist. 

Thus the supremum is defined as the least upper bound of a set of reals, if it exists, or one of negative or positive infinity, which exist at the moment as symbobls only. 

The least upper bound and supremum are a basic property of the reals that does not exist for the rationals. With them, the \( \sqrt{2} \) can be shown to exist as a real.

Upper bound def →least upper bound def→ uniqueness of least upper bound → existence of least upper bound → supremum def


Tao, Analysis I