#Contents# * table of contents {:toc} ## Definition ## Let $F$ be an [[Archimedean ordered field]] and let $$F_{+} \coloneqq \sum_{a:F} 0 \lt a$$ be the positive elements in $F$. $F$ is **sequentially Cauchy complete** if every Cauchy sequence in $F$ converges: $$isCauchy(x) \coloneqq \forall \epsilon \in F_{+}. \exists N \in I. \forall i \in I. \forall j \in I. (i \geq N) \wedge (j \geq N) \wedge (\vert x_i - x_j \vert \lt \epsilon)$$ $$isLimit(x, l) \coloneqq \forall \epsilon \in F_{+}. \exists N \in I. \forall i \in I. (i \geq N) \to (\vert x_i - l \vert \lt \epsilon)$$ $$\forall x \in F^\mathbb{N}. isCauchy(x) \wedge \exists l \in F. isLimit(x, l)$$ ## See also ## * [[Cauchy sequence]] * [[Archimedean ordered field]] category: not redirected to nlab yet