[[!redirects Cauchy sequence]] #Contents# * table of contents {:toc} ## Definition ## ### In premetric spaces ### Let $\mathbb{Q}$ be the [[rational numbers]] and let $$\mathbb{Q}_{+} \coloneqq \sum_{x:\mathbb{Q}} 0 \lt x$$ be the positive rational numbers. Let $S$ be an [[premetric space]]. A [[net]] $x: I \to S$ is a __Cauchy net__ if $$x:I \to S \vdash c(x):\prod_{\epsilon:\mathbb{Q}_{+}} \Vert \sum_{N:I} \prod_{i:I} \prod_{j:I} (i \geq N) \times (j \geq N) \times (x_i \sim_{\epsilon} x_j) \Vert$$ ### Cauchy approximations ### Let $\mathbb{Q}$ be the [[rational numbers]] and let $$\mathbb{Q}_{+} \coloneqq \sum_{x:\mathbb{Q}} 0 \lt x$$ be the positive rational numbers. A net $x: \mathbb{Q}_{+} \to S$ is a __Cauchy approximation__ if $$x:\mathbb{Q}_{+} \to S \vdash c(x):\prod_{\delta:R_{+}} \prod_{\eta:R_{+}} x_\delta \sim_{\delta + \eta} x_\eta$$ Every [[Cauchy approximation]] is a Cauchy net indexed by $\mathbb{Q}_{+}$. This is because $\mathbb{Q}_{+}$ is a strictly ordered type, and thus a directed type and a strictly codirected type, with $N:\mathbb{Q}_{+}$ defined as $N \coloneqq \delta \otimes \eta$ for $\delta:R\mathbb{Q}_{+}$ and $\eta:\mathbb{Q}_{+}$. $\epsilon:\mathbb{Q}_{+}$ is defined as $\epsilon \coloneqq \delta + \eta$. ### In Cauchy spaces ### ... ### Cauchy sequences ### A __Cauchy sequence__ is a Cauchy net whose index type is the [[natural numbers]] $\mathbb{N}$. ## See also ## * [[Cauchy approximation]] * [[Cauchy structure]] * [[premetric space]] * [[net]] * [[filter]]