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