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topology.metric_space.cau_seq_filter

Completeness in terms of cauchy filters vs is_cau_seq sequences #

In this file we apply metric.complete_of_cauchy_seq_tendsto to prove that a normed_ring is complete in terms of cauchy filter if and only if it is complete in terms of cau_seq Cauchy sequences.

theorem cauchy_seq.is_cau_seq {β : Type v} [normed_field β] {f : β} (hf : cauchy_seq f) :

In a normed field, cau_seq coincides with the usual notion of Cauchy sequences.

@[protected, instance]

A complete normed field is complete as a metric space, as Cauchy sequences converge by assumption and this suffices to characterize completeness.