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

Limit points

Let \( (a_n)_{n=m}^{\infty} \) be a sequence of real numbers, let x be a real number, and let \( \epsilon > 0 \) be a real number. We say that \( x \) is ε-adherent to \( (a_n)_{n=m}^{\infty} \) iff there exists an \( n \ge m \) such that [...]. We say that \( x \) is continually ε-adherent to \( (a_n)_{n=m}^{\infty} \) if it is ε-adherent to [...] for every \( N \ge m \). We say that \( x \) is a limit point or adherent point of \( (a_n)_{n=m}^{\infty} \) if it is continually ε-adherent to [...] for every [...].