\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \)
deepdream of
          a sidewalk
Show Question
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)
Math and science::Analysis::Tao::05. The real numbers

Equivalent sequences

Two sequences \( (a_n)_{n=0}^{\infty} \) and \( (b_n)_{n=0}^{\infty} \) are equivalent iff for each rational \( \varepsilon \gt 0 \), the sequences \( (a_n)_{n=0}^{\infty} \) and \( (b_n)_{n=0}^{\infty} \) are eventually ε-close.

In other words the sequences \( a_0, a_1, a_2, ... \) and \( b_0, b_1, b_2, ... \) are equivalent iff for every rational \( \varepsilon \ge 0 \) there exists an \( N \ge 0 \) such that \( |a_k - b_k| \le \varepsilon \) for all \( k \ge N \).


From sequences to reals

sequence → ε-steady sequence → eventually ε-steady sequence → Cauchy sequence → ε-close sequences → eventually ε-close sequences → equivalent sequences → real numbers.

Example

The two sequences

\[ 1.1, 1.01, 1.001, 1.0001, ... \]
and
\[ 0.9, 0.99, 0.999, 0.9999, ... \]
are equivalent.


Source

Tao, Analysis I