Real numbers/Convergent sequence/Cauchy sequence/Fact/Proof

Proof

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a convergent sequence with limit ${}x$. Let ${}\epsilon >0$ be given. We apply the convergence property for ${}\epsilon /2$. Therefore there exists an ${}n_{0}$ with

$\vert {x_{n}-x}\vert \leq \epsilon /2{\text{ for all }}n\geq n_{0}.$

For arbitrary ${}n,m\geq n_{0}$ we then have due to the triangle inequality

${}\vert {x_{n}-x_{m}}\vert \leq \vert {x_{n}-x}\vert +\vert {x-x_{m}}\vert \leq \epsilon /2+\epsilon /2=\epsilon \,.$

  Hence we have a Cauchy sequence.

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