# Real numbers/Bounded monotonic increasing sequence/Cauchy sequence/Fact/Proof

Proof

Let ${\displaystyle {}b\in \mathbb {R} }$ denote a bound from above, so that ${\displaystyle {}x_{n}\leq b}$ holds for all ${\displaystyle {}x_{n}}$. We assume that ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ is not a Cauchy sequence. Then there exists some ${\displaystyle {}\epsilon >0}$ such that for every ${\displaystyle {}n_{0}}$, there exist indices ${\displaystyle {}n>m\geq n_{0}}$ fulfilling ${\displaystyle {}x_{n}-x_{m}\geq \epsilon }$. Because of the monotonicity, there is also for every ${\displaystyle {}n_{0}}$ an ${\displaystyle {}n>n_{0}}$ with ${\displaystyle {}x_{n}-x_{n_{0}}\geq \epsilon }$. Hence, we can define inductively an increasing sequence of natural numbers satisfying

${\displaystyle n_{1}>n_{0}{\text{ such that }}x_{n_{1}}-x_{n_{0}}\geq \epsilon ,}$
${\displaystyle n_{2}>n_{1}{\text{ such that }}x_{n_{2}}-x_{n_{1}}\geq \epsilon ,}$

and so on. On the other hand, there exists, due to the axiom of Archimedes, some ${\displaystyle {}k\in \mathbb {N} }$ with

${\displaystyle {}k\epsilon >b-x_{n_{0}}\,.}$

The sum of the first ${\displaystyle {}k}$ differences of the subsequence ${\displaystyle {}x_{n_{j}}}$, ${\displaystyle {}j\in \mathbb {N} }$, is

{\displaystyle {}{\begin{aligned}x_{n_{k}}-x_{n_{0}}&={\left(x_{n_{k}}-x_{n_{k-1}}\right)}+{\left(x_{n_{k-1}}-x_{n_{k-2}}\right)}+\cdots +{\left(x_{n_{2}}-x_{n_{1}}\right)}+{\left(x_{n_{1}}-x_{n_{0}}\right)}\\&\geq k\epsilon \\&>b-x_{n_{0}}.\end{aligned}}}

This implies ${\displaystyle {}x_{n_{k}}>b}$, contradicting the condition that ${\displaystyle {}b}$ is an upper bound for the sequence.