mathlib documentation

topology.algebra.order.liminf_limsup

Lemmas about liminf and limsup in an order topology. #

theorem filter.tendsto.is_bounded_under_le {α : Type u} {β : Type v} [semilattice_sup α] [topological_space α] [order_topology α] {f : filter β} {u : β α} {a : α} (h : filter.tendsto u f (nhds a)) :
theorem filter.tendsto.is_cobounded_under_ge {α : Type u} {β : Type v} [semilattice_sup α] [topological_space α] [order_topology α] {f : filter β} {u : β α} {a : α} [f.ne_bot] (h : filter.tendsto u f (nhds a)) :
theorem filter.tendsto.is_bounded_under_ge {α : Type u} {β : Type v} [semilattice_inf α] [topological_space α] [order_topology α] {f : filter β} {u : β α} {a : α} (h : filter.tendsto u f (nhds a)) :
theorem filter.tendsto.is_cobounded_under_le {α : Type u} {β : Type v} [semilattice_inf α] [topological_space α] [order_topology α] {f : filter β} {u : β α} {a : α} [f.ne_bot] (h : filter.tendsto u f (nhds a)) :
theorem filter.tendsto.is_bounded_under_ge_at_top {β : Type v} {α : Type u_1} [nonempty α] [preorder α] {f : filter β} {u : β α} (h : filter.tendsto u f filter.at_top) :
theorem lt_mem_sets_of_Limsup_lt {α : Type u} [conditionally_complete_linear_order α] {f : filter α} {b : α} (h : filter.is_bounded has_le.le f) (l : f.Limsup < b) :
∀ᶠ (a : α) in f, a < b
theorem gt_mem_sets_of_Liminf_gt {α : Type u} [conditionally_complete_linear_order α] {f : filter α} {b : α} :
filter.is_bounded ge f b < f.Liminf (∀ᶠ (a : α) in f, b < a)
theorem le_nhds_of_Limsup_eq_Liminf {α : Type u} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] {f : filter α} {a : α} (hl : filter.is_bounded has_le.le f) (hg : filter.is_bounded ge f) (hs : f.Limsup = a) (hi : f.Liminf = a) :
f nhds a

If the liminf and the limsup of a filter coincide, then this filter converges to their common value, at least if the filter is eventually bounded above and below.

theorem Liminf_eq_of_le_nhds {α : Type u} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] {f : filter α} {a : α} [f.ne_bot] (h : f nhds a) :
f.Liminf = a

If a filter is converging, its limsup coincides with its limit.

If a filter is converging, its liminf coincides with its limit.

theorem filter.tendsto.limsup_eq {α : Type u} {β : Type v} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] {f : filter β} {u : β α} {a : α} [f.ne_bot] (h : filter.tendsto u f (nhds a)) :

If a function has a limit, then its limsup coincides with its limit.

theorem filter.tendsto.liminf_eq {α : Type u} {β : Type v} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] {f : filter β} {u : β α} {a : α} [f.ne_bot] (h : filter.tendsto u f (nhds a)) :

If a function has a limit, then its liminf coincides with its limit.

theorem tendsto_of_liminf_eq_limsup {α : Type u} {β : Type v} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] {f : filter β} {u : β α} {a : α} (hinf : filter.liminf u f = a) (hsup : filter.limsup u f = a) (h : filter.is_bounded_under has_le.le f u . "is_bounded_default") (h' : filter.is_bounded_under ge f u . "is_bounded_default") :

If the liminf and the limsup of a function coincide, then the limit of the function exists and has the same value

theorem tendsto_of_le_liminf_of_limsup_le {α : Type u} {β : Type v} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] {f : filter β} {u : β α} {a : α} (hinf : a filter.liminf u f) (hsup : filter.limsup u f a) (h : filter.is_bounded_under has_le.le f u . "is_bounded_default") (h' : filter.is_bounded_under ge f u . "is_bounded_default") :

If a number a is less than or equal to the liminf of a function f at some filter and is greater than or equal to the limsup of f, then f tends to a along this filter.

theorem tendsto_of_no_upcrossings {α : Type u} {β : Type v} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] [densely_ordered α] {f : filter β} {u : β α} {s : set α} (hs : dense s) (H : (a : α), a s (b : α), b s a < b ¬((∃ᶠ (n : β) in f, u n < a) ∃ᶠ (n : β) in f, b < u n)) (h : filter.is_bounded_under has_le.le f u . "is_bounded_default") (h' : filter.is_bounded_under ge f u . "is_bounded_default") :
(c : α), filter.tendsto u f (nhds c)

Assume that, for any a < b, a sequence can not be infinitely many times below a and above b. If it is also ultimately bounded above and below, then it has to converge. This even works if a and b are restricted to a dense subset.

An antitone function between complete linear ordered spaces sends a filter.Limsup to the filter.liminf of the image if it is continuous at the Limsup.

theorem antitone.map_limsup_of_continuous_at {ι : Type u_1} {R : Type u_2} {S : Type u_3} {F : filter ι} [F.ne_bot] [complete_linear_order R] [topological_space R] [order_topology R] [complete_linear_order S] [topological_space S] [order_topology S] {f : R S} (f_decr : antitone f) (a : ι R) (f_cont : continuous_at f (filter.limsup a F)) :

A continuous antitone function between complete linear ordered spaces sends a filter.limsup to the filter.liminf of the images.

An antitone function between complete linear ordered spaces sends a filter.Liminf to the filter.limsup of the image if it is continuous at the Liminf.

theorem antitone.map_liminf_of_continuous_at {ι : Type u_1} {R : Type u_2} {S : Type u_3} {F : filter ι} [F.ne_bot] [complete_linear_order R] [topological_space R] [order_topology R] [complete_linear_order S] [topological_space S] [order_topology S] {f : R S} (f_decr : antitone f) (a : ι R) (f_cont : continuous_at f (filter.liminf a F)) :

A continuous antitone function between complete linear ordered spaces sends a filter.liminf to the filter.limsup of the images.

A monotone function between complete linear ordered spaces sends a filter.Limsup to the filter.limsup of the image if it is continuous at the Limsup.

theorem monotone.map_limsup_of_continuous_at {ι : Type u_1} {R : Type u_2} {S : Type u_3} {F : filter ι} [F.ne_bot] [complete_linear_order R] [topological_space R] [order_topology R] [complete_linear_order S] [topological_space S] [order_topology S] {f : R S} (f_incr : monotone f) (a : ι R) (f_cont : continuous_at f (filter.limsup a F)) :

A continuous monotone function between complete linear ordered spaces sends a filter.limsup to the filter.limsup of the images.

A monotone function between complete linear ordered spaces sends a filter.Liminf to the filter.liminf of the image if it is continuous at the Liminf.

theorem monotone.map_liminf_of_continuous_at {ι : Type u_1} {R : Type u_2} {S : Type u_3} {F : filter ι} [F.ne_bot] [complete_linear_order R] [topological_space R] [order_topology R] [complete_linear_order S] [topological_space S] [order_topology S] {f : R S} (f_incr : monotone f) (a : ι R) (f_cont : continuous_at f (filter.liminf a F)) :

A continuous monotone function between complete linear ordered spaces sends a filter.liminf to the filter.liminf of the images.

theorem infi_eq_of_forall_le_of_tendsto {ι : Type u_1} {R : Type u_2} [complete_linear_order R] [topological_space R] [order_topology R] {x : R} {as : ι R} (x_le : (i : ι), x as i) {F : filter ι} [F.ne_bot] (as_lim : filter.tendsto as F (nhds x)) :
( (i : ι), as i) = x
theorem supr_eq_of_forall_le_of_tendsto {ι : Type u_1} {R : Type u_2} [complete_linear_order R] [topological_space R] [order_topology R] {x : R} {as : ι R} (le_x : (i : ι), as i x) {F : filter ι} [F.ne_bot] (as_lim : filter.tendsto as F (nhds x)) :
( (i : ι), as i) = x
theorem Union_Ici_eq_Ioi_of_lt_of_tendsto {R : Type u_2} [complete_linear_order R] [topological_space R] [order_topology R] {ι : Type u_1} (x : R) {as : ι R} (x_lt : (i : ι), x < as i) {F : filter ι} [F.ne_bot] (as_lim : filter.tendsto as F (nhds x)) :
( (i : ι), set.Ici (as i)) = set.Ioi x
theorem Union_Iic_eq_Iio_of_lt_of_tendsto {R : Type u_2} [complete_linear_order R] [topological_space R] [order_topology R] {ι : Type u_1} (x : R) {as : ι R} (lt_x : (i : ι), as i < x) {F : filter ι} [F.ne_bot] (as_lim : filter.tendsto as F (nhds x)) :
( (i : ι), set.Iic (as i)) = set.Iio x