\(
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\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{"}}
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Math and science::Analysis::Tao::06. Limits of sequences

# Monotone bounded sequences converge

#### Increasing, decreasing and monotone

A sequence \( (a_n)_{n=0}^{\infty} \) is *increasing*
if \( a_n \le a_{n+1} \) for all \( n \in \mathbb{N} \) and *decreasing*
if \( a_n \gt a_{n+1} \) for all \( n \in \mathbb{N} \). A sequence is
*monotone* if it is either increasing or decreasing.

### Monotone bounded sequences converge

If a sequence is [...] and [...], then it converges.

The easiest card ever. The proof is on the reverse; can you think of it? The proof is easy and obvious in retrospect, but it uses a property that I originally didn't consider using to solve the problem.