mathlib documentation

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.

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theorem cau_seq.tendsto_limit {β : Type v} [normed_ring β] [hn : is_absolute_value norm] (f : cau_seq β norm) [cau_seq.is_complete β norm] :
filter.tendsto ⇑f filter.at_top (𝓝 f.lim)

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@[instance]
def normed_field.is_absolute_value {β : Type v} [normed_field β] :
is_absolute_value norm

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

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theorem cau_seq.cauchy_seq {β : Type v} [normed_field β] (f : cau_seq β norm) :
cauchy_seq ⇑f

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theorem cau_seq_iff_cauchy_seq {α : Type u} [normed_field α] {u : ℕ → α} :
is_cau_seq norm u ↔ cauchy_seq u

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

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@[instance]
def complete_space_of_cau_seq_complete {β : Type v} [normed_field β] [cau_seq.is_complete β norm] :
complete_space β

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