\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \)
header
Show Answer
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \)
Math and science::Analysis::Tao::05. The real numbers

Bounded sequences

Let \( M >0 \) be a rational. 

A finite sequence (of rationals, but equally applies to reals) \( a_1, a_2, a_3, ..., a_n \) is bounded by M iff [...] for all \( 1 \ge i \ge n  \).

An infinite sequence \( (a_n)_{n=1}^{\infty} \) is bounded by M iff [...] for all \( i \ge 1 \).

A sequence is said to be bounded iff it is bounded by some \( M \ge 0 \).