\(
\newcommand{\cat}[1] {\mathrm{#1}}
\newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})}
\newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}}
\newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}}
\newcommand{\betaReduction}[0] {\rightarrow_{\beta}}
\newcommand{\betaEq}[0] {=_{\beta}}
\newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}}
\newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}}
\newcommand{\groupMul}[1] { \cdot_{\small{#1}}}
\newcommand{\groupAdd}[1] { +_{\small{#1}}}
\newcommand{\inv}[1] {#1^{-1} }
\newcommand{\bm}[1] { \boldsymbol{#1} }
\newcommand{\qed} { {\scriptstyle \Box} }
\require{physics}
\require{ams}
\require{mathtools}
\)
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:
- \( M \) is an upper bound for \( E \).
- 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
