\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)

Math and science::Analysis::Tao::06. Limits of sequences

Let \( (a_n)_{n=m}^{\infty} \) be a sequence of real numbers. Then we define \( \sup(a_n)_{n=m}^{\infty} \) to be the supremum of the set \( \{a_n : n \ge m\} \), and \( \inf(a_n)_{n=m}^{\infty} \) to be the infimum of the same set \( \{a_n : n \ge m\} \).

The quantities \( \sup(a_n)_{n=m}^{\infty} \) and \( \inf(a_n)_{n=m}^{\infty} \) are sometimes written as \( \sup_{n \ge m}a_n \) and \( \inf_{n \ge m}a_n \) respectively.

Sequences as functions

When sequences are viewed as functions from \( \mathbb{N} \) to \( \mathbb{R} \), the supremum of the sequence can be viewed as the supremum of the codomain of the function.