Real numbers/Unit fractions/Example

The sequence

${}x_{n}={\frac {1}{n}}\,$

converges to the limit ${}0$. To show this, let some positive ${}\epsilon$ be given. Due to the Archimedean axiom, there exists an ${}n_{0}$, such that ${}{\frac {1}{n_{0}}}\leq \epsilon$. Then for all ${}n\geq n_{0}$, the estimate

${}\vert {x_{n}-0}\vert ={\frac {1}{n}}\leq {\frac {1}{n_{0}}}\leq \epsilon \,$

holds.