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Theorem List for Metamath Proof Explorer - 41901-42000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsupcnvlimsup 41901* If a function on a set of upper integers has a real superior limit, the supremum of the rightmost parts of the function, converges to that superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim sup‘𝐹) ∈ ℝ)       (𝜑 → (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )) ⇝ (lim sup‘𝐹))
 
Theoremsupcnvlimsupmpt 41902* If a function on a set of upper integers has a real superior limit, the supremum of the rightmost parts of the function, converges to that superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑗𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (lim sup‘(𝑗𝑍𝐵)) ∈ ℝ)       (𝜑 → (𝑘𝑍 ↦ sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < )) ⇝ (lim sup‘(𝑗𝑍𝐵)))
 
Theorem0cnv 41903 If (/) is a complex number, then it converges to itself. (see 0ncn 10544 and its comment ; see also the comment in climlimsupcex 41930) (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(∅ ∈ ℂ → ∅ ⇝ ∅)
 
Theoremclimuzlem 41904* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℂ)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
 
Theoremclimuz 41905* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℂ)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
 
Theoremlmbr3v 41906* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐽 ∈ (TopOn‘𝑋))       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))
 
Theoremclimisp 41907* If a sequence converges to an isolated point (w.r.t. the standard topology on the complex numbers) then the sequence eventually becomes that point. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℂ)    &   (𝜑𝐹𝐴)    &   (𝜑𝑋 ∈ ℝ+)    &   ((𝜑𝑘𝑍 ∧ (𝐹𝑘) ≠ 𝐴) → 𝑋 ≤ (abs‘((𝐹𝑘) − 𝐴)))       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) = 𝐴)
 
Theoremlmbr3 41908* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝐹    &   (𝜑𝐽 ∈ (TopOn‘𝑋))       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))
 
Theoremclimrescn 41909* A sequence converging w.r.t. the standard topology on the complex numbers, eventually becomes a sequence of complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹 Fn 𝑍)    &   (𝜑𝐹 ∈ dom ⇝ )       (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℂ)
 
Theoremclimxrrelem 41910* If a seqence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹𝐴)    &   (𝜑𝐷 ∈ ℝ+)    &   ((𝜑 ∧ +∞ ∈ ℂ) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))    &   ((𝜑 ∧ -∞ ∈ ℂ) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))       (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
 
Theoremclimxrre 41911* If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued (the weaker hypothesis 𝐹 ∈ dom ⇝ is probably not enough, since in principle we could have +∞ ∈ ℂ and -∞ ∈ ℂ). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐹𝐴)       (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
 
20.36.7.1  Inferior limit (lim inf)
 
Syntaxclsi 41912 Extend class notation to include the liminf function. (actually, it makes sense for any extended real function defined on a subset of RR which is not upper-bounded)
class lim inf
 
Definitiondf-liminf 41913* Define the inferior limit of a sequence of extended real numbers. (Contributed by GS, 2-Jan-2022.)
lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
 
Theoremlimsuplt2 41914* The defining property of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐹:𝐵⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ*)       (𝜑 → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑘 ∈ ℝ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴))
 
Theoremliminfgord 41915 Ordering property of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → inf(((𝐹 “ (𝐴[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝐵[,)+∞)) ∩ ℝ*), ℝ*, < ))
 
Theoremlimsupvald 41916* The superior limit of a sequence 𝐹 of extended real numbers is the infimum of the set of suprema of all restrictions of 𝐹 to an upperset of reals . (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
 
Theoremlimsupresicompt 41917* The superior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℝ)    &   𝑍 = (𝑀[,)+∞)       (𝜑 → (lim sup‘(𝑥𝐴𝐵)) = (lim sup‘(𝑥 ∈ (𝐴𝑍) ↦ 𝐵)))
 
Theoremlimsupcli 41918 Closure of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐹𝑉       (lim sup‘𝐹) ∈ ℝ*
 
Theoremliminfgf 41919 Closure of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       𝐺:ℝ⟶ℝ*
 
Theoremliminfval 41920* The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝐹𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
 
Theoremclimlimsup 41921 A sequence of real numbers converges if and only if it converges to its superior limit. The first hypothesis is needed (see climlimsupcex 41930 for a counterexample) (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim sup‘𝐹)))
 
Theoremlimsupge 41922* The defining property of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐹:𝐵⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ*)       (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑘 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )))
 
Theoremliminfgval 41923* Value of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝑀 ∈ ℝ → (𝐺𝑀) = inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < ))
 
Theoremliminfcl 41924 Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐹𝑉 → (lim inf‘𝐹) ∈ ℝ*)
 
Theoremliminfvald 41925* The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
 
Theoremliminfval5 41926* The inferior limit of an infinite sequence 𝐹 of extended real numbers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   𝐺 = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))       (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
 
Theoremlimsupresxr 41927 The superior limit of a function only depends on the restriction of that function to the preimage of the set of extended reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → Fun 𝐹)    &   𝐴 = (𝐹 “ ℝ*)       (𝜑 → (lim sup‘(𝐹𝐴)) = (lim sup‘𝐹))
 
Theoremliminfresxr 41928 The inferior limit of a function only depends on the preimage of the extended real part. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → Fun 𝐹)    &   𝐴 = (𝐹 “ ℝ*)       (𝜑 → (lim inf‘(𝐹𝐴)) = (lim inf‘𝐹))
 
Theoremliminfval2 41929* The superior limit, relativized to an unbounded set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    &   (𝜑𝐹𝑉)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)       (𝜑 → (lim inf‘𝐹) = sup((𝐺𝐴), ℝ*, < ))
 
Theoremclimlimsupcex 41930 Counterexample for climlimsup 41921, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 10544 and its comment) (Contributed by Glauco Siliprandi, 2-Jan-2022.)
¬ 𝑀 ∈ ℤ    &   𝑍 = (ℤ𝑀)    &   𝐹 = ∅       ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹:𝑍⟶ℝ ∧ 𝐹 ∈ dom ⇝ ∧ ¬ 𝐹 ⇝ (lim sup‘𝐹)))
 
Theoremliminfcld 41931 Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)       (𝜑 → (lim inf‘𝐹) ∈ ℝ*)
 
Theoremliminfresico 41932 The inferior limit doesn't change when a function is restricted to an upperset of reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℝ)    &   𝑍 = (𝑀[,)+∞)    &   (𝜑𝐹𝑉)       (𝜑 → (lim inf‘(𝐹𝑍)) = (lim inf‘𝐹))
 
Theoremlimsup10exlem 41933* The range of the given function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))    &   (𝜑𝐾 ∈ ℝ)       (𝜑 → (𝐹 “ (𝐾[,)+∞)) = {0, 1})
 
Theoremlimsup10ex 41934 The superior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))       (lim sup‘𝐹) = 1
 
Theoremliminf10ex 41935 The inferior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))       (lim inf‘𝐹) = 0
 
Theoremliminflelimsuplem 41936* The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)       (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
 
Theoremliminflelimsup 41937* The superior limit is greater than or equal to the inferior limit. The second hypothesis is needed (see liminflelimsupcex 41958 for a counterexample). The inequality can be strict, see liminfltlimsupex 41942. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)       (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
 
Theoremlimsupgtlem 41938* For any positive real, the superior limit of F is larger than any of its values at large enough arguments, up to that positive real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim sup‘𝐹) ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − 𝑋) < (lim sup‘𝐹))
 
Theoremlimsupgt 41939* Given a sequence of real numbers, there exists an upper part of the sequence that's appxoximated from below by the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim sup‘𝐹) ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − 𝑋) < (lim sup‘𝐹))
 
Theoremliminfresre 41940 The inferior limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)       (𝜑 → (lim inf‘(𝐹 ↾ ℝ)) = (lim inf‘𝐹))
 
Theoremliminfresicompt 41941* The inferior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℝ)    &   𝑍 = (𝑀[,)+∞)    &   (𝜑𝐴𝑉)       (𝜑 → (lim inf‘(𝑥 ∈ (𝐴𝑍) ↦ 𝐵)) = (lim inf‘(𝑥𝐴𝐵)))
 
Theoremliminfltlimsupex 41942 An example where the lim inf is strictly smaller than the lim sup. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))       (lim inf‘𝐹) < (lim sup‘𝐹)
 
Theoremliminfgelimsup 41943* The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)       (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
 
Theoremliminfvalxr 41944* Alternate definition of lim inf when 𝐹 is an extended real-valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝐹    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶ℝ*)       (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑥𝐴 ↦ -𝑒(𝐹𝑥))))
 
Theoremliminfresuz 41945 If the real part of the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ)       (𝜑 → (lim inf‘(𝐹𝑍)) = (lim inf‘𝐹))
 
Theoremliminflelimsupuz 41946 The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
 
Theoremliminfvalxrmpt 41947* Alternate definition of lim inf when 𝐹 is an extended real-valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)       (𝜑 → (lim inf‘(𝑥𝐴𝐵)) = -𝑒(lim sup‘(𝑥𝐴 ↦ -𝑒𝐵)))
 
Theoremliminfresuz2 41948 If the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑 → dom 𝐹 ⊆ ℤ)       (𝜑 → (lim inf‘(𝐹𝑍)) = (lim inf‘𝐹))
 
Theoremliminfgelimsupuz 41949 The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
 
Theoremliminfval4 41950* Alternate definition of lim inf when the given function is eventually real-valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ)       (𝜑 → (lim inf‘(𝑥𝐴𝐵)) = -𝑒(lim sup‘(𝑥𝐴 ↦ -𝐵)))
 
Theoremliminfval3 41951* Alternate definition of lim inf when the given function is eventually extended real-valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*)       (𝜑 → (lim inf‘(𝑥𝐴𝐵)) = -𝑒(lim sup‘(𝑥𝐴 ↦ -𝑒𝐵)))
 
Theoremliminfequzmpt2 41952* Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑗𝜑    &   𝑗𝐴    &   𝑗𝐵    &   𝐴 = (ℤ𝑀)    &   𝐵 = (ℤ𝑁)    &   (𝜑𝐾𝐴)    &   (𝜑𝐾𝐵)    &   ((𝜑𝑗 ∈ (ℤ𝐾)) → 𝐶𝑉)       (𝜑 → (lim inf‘(𝑗𝐴𝐶)) = (lim inf‘(𝑗𝐵𝐶)))
 
Theoremliminfvaluz 41953* Alternate definition of lim inf for an extended real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ*)       (𝜑 → (lim inf‘(𝑘𝑍𝐵)) = -𝑒(lim sup‘(𝑘𝑍 ↦ -𝑒𝐵)))
 
Theoremliminf0 41954 The inferior limit of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(lim inf‘∅) = +∞
 
Theoremlimsupval4 41955* Alternate definition of lim inf when the given a function is eventually extended real-valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*)       (𝜑 → (lim sup‘(𝑥𝐴𝐵)) = -𝑒(lim inf‘(𝑥𝐴 ↦ -𝑒𝐵)))
 
Theoremliminfvaluz2 41956* Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)       (𝜑 → (lim inf‘(𝑘𝑍𝐵)) = -𝑒(lim sup‘(𝑘𝑍 ↦ -𝐵)))
 
Theoremliminfvaluz3 41957* Alternate definition of lim inf for an extended real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘𝑍 ↦ -𝑒(𝐹𝑘))))
 
Theoremliminflelimsupcex 41958 A counterexample for liminflelimsup 41937, showing that the second hypothesis is needed. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(lim sup‘∅) < (lim inf‘∅)
 
Theoremlimsupvaluz3 41959* Alternate definition of lim inf for an extended real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ*)       (𝜑 → (lim sup‘(𝑘𝑍𝐵)) = -𝑒(lim inf‘(𝑘𝑍 ↦ -𝑒𝐵)))
 
Theoremliminfvaluz4 41960* Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘𝑍 ↦ -(𝐹𝑘))))
 
Theoremlimsupvaluz4 41961* Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)       (𝜑 → (lim sup‘(𝑘𝑍𝐵)) = -𝑒(lim inf‘(𝑘𝑍 ↦ -𝐵)))
 
Theoremclimliminflimsupd 41962 If a sequence of real numbers converges, its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑𝐹 ∈ dom ⇝ )       (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹))
 
Theoremliminfreuzlem 41963* Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
 
Theoremliminfreuz 41964* Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
 
Theoremliminfltlem 41965* Given a sequence of real numbers, there exists an upper part of the sequence that's approximated from above by the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim inf‘𝐹) ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(lim inf‘𝐹) < ((𝐹𝑘) + 𝑋))
 
Theoremliminflt 41966* Given a sequence of real numbers, there exists an upper part of the sequence that's approximated from above by the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim inf‘𝐹) ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(lim inf‘𝐹) < ((𝐹𝑘) + 𝑋))
 
Theoremclimliminf 41967 A sequence of real numbers converges if and only if it converges to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim inf‘𝐹)))
 
Theoremliminflimsupclim 41968 A sequence of real numbers converges if its inferior limit is real, and it is greater than or equal to the superior limit (in such a case, they are actually equal, see liminflelimsupuz 41946). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim inf‘𝐹) ∈ ℝ)    &   (𝜑 → (lim sup‘𝐹) ≤ (lim inf‘𝐹))       (𝜑𝐹 ∈ dom ⇝ )
 
Theoremclimliminflimsup 41969 A sequence of real numbers converges if and only if its inferior limit is real and it is greater than or equal to its superior limit (in such a case, they are actually equal, see liminfgelimsupuz 41949). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
 
Theoremclimliminflimsup2 41970 A sequence of real numbers converges if and only if its superior limit is real and it is less than or equal to its inferior limit (in such a case, they are actually equal, see liminfgelimsupuz 41949). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
 
Theoremclimliminflimsup3 41971 A sequence of real numbers converges if and only if its inferior limit is real and equal to its superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim inf‘𝐹) = (lim sup‘𝐹))))
 
Theoremclimliminflimsup4 41972 A sequence of real numbers converges if and only if its superior limit is real and equal to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim inf‘𝐹) = (lim sup‘𝐹))))
 
Theoremlimsupub2 41973* A extended real valued function, with limsup that is not +∞, is eventually less than +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
𝑗𝜑    &   𝑗𝐹    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → (lim sup‘𝐹) ≠ +∞)       (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < +∞))
 
Theoremlimsupubuz2 41974* A sequence with values in the extended reals, and with limsup that is not +∞, is eventually less than +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
𝑗𝜑    &   𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑 → (lim sup‘𝐹) ≠ +∞)       (𝜑 → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) < +∞)
 
Theoremxlimpnfxnegmnf 41975* A sequence converges to +∞ if and only if its negation converges to -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
𝑗𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥))
 
Theoremliminflbuz2 41976* A sequence with values in the extended reals, and with liminf that is not -∞, is eventually greater than -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
𝑗𝜑    &   𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑 → (lim inf‘𝐹) ≠ -∞)       (𝜑 → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)-∞ < (𝐹𝑗))
 
Theoremliminfpnfuz 41977* The inferior limit of a function is +∞ if and only if every real number is the lower bound of the restriction of the function to a set of upper integers. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → ((lim inf‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
 
Theoremliminflimsupxrre 41978* A sequence with values in the extended reals, and with real liminf and limsup, is eventually real. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑 → (lim sup‘𝐹) ≠ +∞)    &   (𝜑 → (lim inf‘𝐹) ≠ -∞)       (𝜑 → ∃𝑘𝑍 (𝐹 ↾ (ℤ𝑘)):(ℤ𝑘)⟶ℝ)
 
20.36.7.2  Limits for sequences of extended real numbers

Textbooks generally use a single symbol to denote the limit of a sequence of real numbers. But then, three distinct definitions are usually given: one for the case of convergence to a real number, one for the case of limit to +∞ and one for the case of limit to -∞. It turns out that these three definitions can be expressed as the limit w.r.t. to the standard topology on the extended reals. In this section, a relation ~~>* is defined that captures all three definitions (and can be applied to sequences of extended reals, also), see dfxlim2 42009.

 
Syntaxclsxlim 41979 Extend class notation with convergence relation for limits in the extended real numbers.
class ~~>*
 
Definitiondf-xlim 41980 Define the convergence relation for extended real sequences. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
~~>* = (⇝𝑡‘(ordTop‘ ≤ ))
 
Theoremxlimrel 41981 The limit on extended reals is a relation. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Rel ~~>*
 
Theoremxlimres 41982 A function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐹 ∈ (ℝ*pm ℂ))    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ𝑀))~~>*𝐴))
 
Theoremxlimcl 41983 The limit of a sequence of extended real numbers is an extended real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝐹~~>*𝐴𝐴 ∈ ℝ*)
 
Theoremrexlimddv2 41984* Restricted existential elimination rule of natural deduction. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑 → ∃𝑥𝐴 𝜓)    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       (𝜑𝜒)
 
Theoremxlimclim 41985 Given a sequence of reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals, if and only if it converges to 𝐴 w.r.t. to the standard topology on the extended reals (see climreeq 41774). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐹~~>*𝐴𝐹𝐴))
 
Theoremxlimconst 41986* A constant sequence converges to its value, w.r.t. the standard topology on the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹 Fn 𝑍)    &   (𝜑𝐴 ∈ ℝ*)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)       (𝜑𝐹~~>*𝐴)
 
Theoremclimxlim 41987 A converging sequence in the reals is a converging sequence in the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑𝐹𝐴)       (𝜑𝐹~~>*𝐴)
 
Theoremxlimbr 41988* Express the binary relation "sequence 𝐹 converges to point 𝑃 " w.r.t. the standard topology on the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   𝐽 = (ordTop‘ ≤ )       (𝜑 → (𝐹~~>*𝑃 ↔ (𝑃 ∈ ℝ* ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))
 
Theoremfuzxrpmcn 41989 A function mapping from an upper set of integers to the extended reals is a partial map on the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑𝐹 ∈ (ℝ*pm ℂ))
 
Theoremcnrefiisplem 41990* Lemma for cnrefiisp 41991 (some local definitions are used). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → ¬ 𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ Fin)    &   𝐶 = (ℝ ∪ 𝐵)    &   𝐷 = ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))})    &   𝑋 = inf(𝐷, ℝ*, < )       (𝜑 → ∃𝑥 ∈ ℝ+𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))))
 
Theoremcnrefiisp 41991* A non-real, complex number is an isolated point w.r.t. the union of the reals with any finite set (the extended reals is an example of such a union). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → ¬ 𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ Fin)    &   𝐶 = (ℝ ∪ 𝐵)       (𝜑 → ∃𝑥 ∈ ℝ+𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))))
 
Theoremxlimxrre 41992* If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued (the weaker hypothesis 𝐹 ∈ dom ⇝ is probably not enough, since in principle we could have +∞ ∈ ℂ and -∞ ∈ ℂ). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐹~~>*𝐴)       (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
 
Theoremxlimmnfvlem1 41993* Lemma for xlimmnfv 41995: the "only if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹~~>*-∞)    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑋)
 
Theoremxlimmnfvlem2 41994* Lemma for xlimmnf 42002: the "if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝜑    &   𝑗𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) < 𝑥)       (𝜑𝐹~~>*-∞)
 
Theoremxlimmnfv 41995* A function converges to minus infinity if it eventually becomes (and stays) smaller than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑥))
 
Theoremxlimconst2 41996* A sequence that eventually becomes constant, converges to its constant value (w.r.t. the standard topology on the extended reals). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝑁𝑍)    &   (𝜑𝐴 ∈ ℝ*)    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐹𝑘) = 𝐴)       (𝜑𝐹~~>*𝐴)
 
Theoremxlimpnfvlem1 41997* Lemma for xlimpnfv 41999: the "only if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹~~>*+∞)    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑋 ≤ (𝐹𝑘))
 
Theoremxlimpnfvlem2 41998* Lemma for xlimpnfv 41999: the "if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝜑    &   𝑗𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 < (𝐹𝑘))       (𝜑𝐹~~>*+∞)
 
Theoremxlimpnfv 41999* A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘)))
 
Theoremxlimclim2lem 42000* Lemma for xlimclim2 42001. Here it is additionally assumed that the sequence will eventually become (and stay) real. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)       (𝜑 → (𝐹~~>*𝐴𝐹𝐴))
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