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

Proof

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

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

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

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

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

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

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

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

To fact