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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.

@[protected, instance]
theorem cauchy_seq.is_cau_seq {β : Type v} [normed_field β] {f : → β} (hf : cauchy_seq f) :
theorem cau_seq.cauchy_seq {β : Type v} [normed_field β] (f : cau_seq β has_norm.norm) :
theorem cau_seq_iff_cauchy_seq {α : Type u} [normed_field α] {u : → α} :

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.