# Real numbers/Sequences/Bounded increasing/Convergence/Decimal expansion/Section

## Corollary

A bounded and monotone sequence in ${\displaystyle {}\mathbb {R} }$ converges.

### Proof

Due to the condition, the sequence is increasing and bounded from above or decreasing and bounded from below. Because of fact, we have a Cauchy sequence which converges in ${\displaystyle {}\mathbb {R} }$.

${\displaystyle \Box }$

This statement is also the reason that any decimal expansion defines a real number. An (infinite) decimal expansion

${\displaystyle a.a_{-1}a_{-2}a_{-3}\ldots }$

with ${\displaystyle {}a\in \mathbb {N} }$ (we restrict to nonnegative numbers) and ${\displaystyle {}a_{-n}\in \{0,\ldots ,9\}}$ is just the sequence of rational numbers

${\displaystyle x_{0}:=a,\,x_{1}:=a+a_{-1}\cdot {\frac {1}{10}},\,x_{2}:=a+a_{-1}\cdot {\frac {1}{10}}+a_{-2}\cdot {\left({\frac {1}{10}}\right)}^{2},\,{\rm {{etc}.}}}$

This sequence is increasing. It is also bounded, e.g. by ${\displaystyle {}a+1}$, so that it defines a Cauchy sequence and thus a real number.