\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \)
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

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 \( |a_i| \le M \) for all \( 1 \ge i \ge n  \).

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

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

Note how the definition is using only a sequence beginning at index 1 (the sequences of form \( (a_n)_{n=1}^{\infty} \) as opposed to the more general form \( (a_n)_{n=m}^{\infty} \) for some integer m). Tao mentions earlier in the text that the beginning index is irrelevant.

Two propositions/lemmas that follow:
1. All finite sequences are bounded. 
2. All Cauchy sequences are bounded. (proof: exercise 5.1.1). 


Tao, Analysis I