[[!redirects Cauchy sequence]] #Contents# * table of contents {:toc} ## Definition ## ### In premetric spaces ### Let $T$ be a [[directed type]], and let $S$ be a $T$-[[premetric space]]. Given a [[directed type]] $I$, a net $x: I \to S$ is a __Cauchy net__ if $$x:I \to S \vdash c(x):\prod_{\epsilon:T} \Vert \sum_{N:I} \prod_{i:I} \prod_{j:I} (i \geq N) \times (j \geq N) \times (x_i \sim_{\epsilon} x_j) \Vert$$ ### Cauchy approximations ### Let $R$ be a [[Archimedean ordered integral domain]] with a [[dense]] [[strict order]], and let $R_{+}$ be the [[semiring]] of positive terms in $R$. If both $T$ and $I$ are $R_{+}$, then a net $x: R_{+} \to S$ is a __Cauchy approximation__ if $$x:R_{+} \to S \vdash c(x):\prod_{\delta:R_{+}} \prod_{\eta:R_{+}} x_\delta \sim_{\delta + \eta} x_\eta$$ ### 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]]