\(
\newcommand{\cat}[1] {\mathrm{#1}}
\newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})}
\newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}}
\newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}}
\newcommand{\betaReduction}[0] {\rightarrow_{\beta}}
\newcommand{\betaEq}[0] {=_{\beta}}
\newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}}
\newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}}
\newcommand{\groupMul}[1] { \cdot_{\small{#1}}}
\newcommand{\groupAdd}[1] { +_{\small{#1}}}
\newcommand{\inv}[1] {#1^{-1} }
\newcommand{\bm}[1] { \boldsymbol{#1} }
\require{physics}
\require{ams}
\require{mathtools}
\)
Math and science::Analysis::Tao::09. Continuous functions on R
Bounded sets (of reals)
A subset \( X \) of the real line is said to be
bounded if
we have
\( X \subseteq [-M, M] \) for some real number \( M > 0 \).
Not to be confused with bounded sequences.
Example
The interval \( [a, b] \) is bounded for any real numbers \( a \) and \( b \), as it is contained inside \( [-M, M] \) where \( M := max(|a|, |b|) \).
The half-infinite intervals and the doubly infinite interval are not bounded.
The sets \( \mathbb{N}, \mathbb{Z}, \mathbb{Q} \text{ and } \mathbb{R} \) are all unbounded.
Source
p217
