\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \)
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

Least upper bound

Let \( E \) be a subset of \( \mathbb{R} \) and let \( M \) be a real number. We say that \( M \) is a least upper bound for E iff:
  1. \( M \) is an upper bound for \( E \).
  2. Any other upper bound for \( E \) is greater or equal to \( M \).

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

Example

The interval \( E:= \{ x \in R : 0 \le x \le 1 \} \) has 1 as a least upper bound. 


The empty set does not have any least upper bound (why?).

Source

Tao, Analysis I