File: Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean

full_name: NonarchimedeanGroup.cauchySeq_prod_of_tendsto_cofinite_one
code: /-- Let `G` be a nonarchimedean multiplicative abelian group, and let `f : α → G` be a function that
tends to one on the filter of cofinite sets. For each finite subset of `α`, consider the partial
product of `f` on that subset. These partial products form a Cauchy filter. -/
@[to_additive "Let `G` be a nonarchimedean additive abelian group, and let `f : α → G` be a function
that tends to zero on the filter of cofinite sets. For each finite subset of `α`, consider the
partial sum of `f` on that subset. These partial sums form a Cauchy filter."]
theorem cauchySeq_prod_of_tendsto_cofinite_one {f : α → G} (hf : Tendsto f cofinite (𝓝 1)) :
    CauchySeq (fun s ↦ ∏ i ∈ s, f i) 
goal: CauchySeq fun s => ∏ i ∈ s, f i

full_name: NonarchimedeanAddGroup.cauchySeq_sum_of_tendsto_cofinite_zero
code: /-- Let `G` be a nonarchimedean multiplicative abelian group, and let `f : α → G` be a function that
tends to one on the filter of cofinite sets. For each finite subset of `α`, consider the partial
product of `f` on that subset. These partial products form a Cauchy filter. -/
@[to_additive "Let `G` be a nonarchimedean additive abelian group, and let `f : α → G` be a function
that tends to zero on the filter of cofinite sets. For each finite subset of `α`, consider the
partial sum of `f` on that subset. These partial sums form a Cauchy filter."]
theorem cauchySeq_prod_of_tendsto_cofinite_one {f : α → G} (hf : Tendsto f cofinite (𝓝 1)) :
    CauchySeq (fun s ↦ ∏ i ∈ s, f i) 
goal: CauchySeq fun s => ∑ i ∈ s, f i

full_name: NonarchimedeanGroup.multipliable_of_tendsto_cofinite_one
code: /-- Let `G` be a complete nonarchimedean multiplicative abelian group, and let `f : α → G` be a
function that tends to one on the filter of cofinite sets. Then `f` is unconditionally
multipliable. -/
@[to_additive "Let `G` be a complete nonarchimedean additive abelian group, and let `f : α → G` be a
function that tends to zero on the filter of cofinite sets. Then `f` is unconditionally summable."]
theorem multipliable_of_tendsto_cofinite_one [CompleteSpace G] {f : α → G}
    (hf : Tendsto f cofinite (𝓝 1)) : Multipliable f 
goal: Multipliable f

full_name: NonarchimedeanAddGroup.summable_of_tendsto_cofinite_zero
code: /-- Let `G` be a complete nonarchimedean multiplicative abelian group, and let `f : α → G` be a
function that tends to one on the filter of cofinite sets. Then `f` is unconditionally
multipliable. -/
@[to_additive "Let `G` be a complete nonarchimedean additive abelian group, and let `f : α → G` be a
function that tends to zero on the filter of cofinite sets. Then `f` is unconditionally summable."]
theorem multipliable_of_tendsto_cofinite_one [CompleteSpace G] {f : α → G}
    (hf : Tendsto f cofinite (𝓝 1)) : Multipliable f 
goal: Summable f

full_name: NonarchimedeanGroup.multipliable_iff_tendsto_cofinite_one
code: /-- Let `G` be a complete nonarchimedean multiplicative abelian group. Then a function `f : α → G`
is unconditionally multipliable if and only if it tends to one on the filter of cofinite sets. -/
@[to_additive "Let `G` be a complete nonarchimedean additive abelian group. Then a function
`f : α → G` is unconditionally summable if and only if it tends to zero on the filter of cofinite
sets."]
theorem multipliable_iff_tendsto_cofinite_one [CompleteSpace G] (f : α → G) :
    Multipliable f ↔ Tendsto f cofinite (𝓝 1) 
goal: Multipliable f ↔ Filter.Tendsto f Filter.cofinite (𝓝 1)

full_name: NonarchimedeanAddGroup.summable_iff_tendsto_cofinite_zero
code: /-- Let `G` be a complete nonarchimedean multiplicative abelian group. Then a function `f : α → G`
is unconditionally multipliable if and only if it tends to one on the filter of cofinite sets. -/
@[to_additive "Let `G` be a complete nonarchimedean additive abelian group. Then a function
`f : α → G` is unconditionally summable if and only if it tends to zero on the filter of cofinite
sets."]
theorem multipliable_iff_tendsto_cofinite_one [CompleteSpace G] (f : α → G) :
    Multipliable f ↔ Tendsto f cofinite (𝓝 1) 
goal: Summable f ↔ Filter.Tendsto f Filter.cofinite (𝓝 0)