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Definition

In premetric spaces

Let $T$ be a directed type, and let $S$ be a $T$-premetric space. Given a directed type $I$, a limit of a net $x: I \to S$ is a term $l:S$ with

Of Cauchy approximations

Let $\mathrm{<span><del\; class="diffmod">\; R</del><ins\; class="diffmod">\; T</ins></span>}$ R T be a Archimedean directed ordered type integral domain with and a dense codirected type where the directed type operationstrict order$\oplus$ , and is let associative. A limit of a$R_{+}T$ R_{+} T - be thesemiring? of positive terms in $R$. If both $T$ and $I$ are $R_{+}$, then a limit of a Cauchy approximation $x:R_{+}T\to S$ x: R_{+} T \to S is a term $l:S$ with

In convergence spaces

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Sequences

A limit of a sequence is a limit of a net that happens to be a sequence.

See also

• premetric space
• modulus of Cauchy convergence
• Cauchy structure
• Cauchy approximation
• HoTT book real numbers