\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)

Subsequences, definition

Let \( (a_n)_{n=0}^{\infty} \) and \( (b_n)_{n=0}^{\infty} \) be sequences of reals. We say that \( (b_n)_{n=m}^{\infty} \) is a subsequence of \( (a_n)_{n=m}^{\infty} \) if there exists a function \( f: \mathbb{N} \rightarrow \mathbb{N} \) which is [...] such that

Given how sequences were originally defined in terms of functions, a more explicit definition would be:

Let \( (a_n)_{n=0}^{\infty} \) be the sequence represented by the function \( g: \mathbb{N} \rightarrow \mathbb{R} \). If \( f: \mathbb{N} \rightarrow \mathbb{N} \) is [...], then the sequence defined by [...] is a subsequence of \( (a_n)_{n=0}^{\infty} \).