Real numbers/Convergent sequence/Bounded/Fact/Proof

Proof

Let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ be the convergent sequence with ${\displaystyle {}x\in \mathbb {R} }$ as its limit. Choose some ${\displaystyle {}\epsilon >0}$. Due to convergence there exists some ${\displaystyle {}n_{0}}$ such that

${\displaystyle \vert {x_{n}-x}\vert \leq \epsilon {\text{ for all }}n\geq n_{0}.}$

So in particular

${\displaystyle \vert {x_{n}}\vert \leq \vert {x}\vert +\vert {x-x_{n}}\vert \leq \vert {x}\vert +\epsilon {\text{ for all }}n\geq n_{0}.}$

Below ${\displaystyle {}n_{0}}$ there are ony finitely many members, hence the maximum

${\displaystyle {}B:=\max _{n

is welldefined. Therefore ${\displaystyle {}B}$ is an upper bound and ${\displaystyle {}-B}$ is a lower bound for ${\displaystyle {}{\left\{x_{n}\mid n\in \mathbb {N} \right\}}}$.