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Theorem List for Metamath Proof Explorer - 40701-40800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfnlimfvre 40701* The limit function of real functions, applied to elements in its domain, evaluates to Real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑚𝜑    &   𝑚𝐹    &   𝑥𝐹    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   (𝜑𝑋𝐷)       (𝜑 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)

Theoremallbutfifvre 40702* Given a sequence of real-valued functions, and 𝑋 that belongs to all but finitely many domains, then its function value is ultimately a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑚𝜑    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)    &   𝐷 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)    &   (𝜑𝑋𝐷)       (𝜑 → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ∈ ℝ)

Theoremclimleltrp 40703* The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐹𝑘) ∈ ℝ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐶)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℝ ∧ (𝐹𝑘) < (𝐶 + 𝑋)))

Theoremfnlimfvre2 40704* The limit function of real functions, applied to elements in its domain, evaluates to Real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑚𝜑    &   𝑚𝐹    &   𝑥𝐹    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝑋𝐷)       (𝜑 → (𝐺𝑋) ∈ ℝ)

Theoremfnlimf 40705* The limit function of real functions, is a real-valued function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑚𝜑    &   𝑚𝐹    &   𝑥𝐹    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))       (𝜑𝐺:𝐷⟶ℝ)

Theoremfnlimabslt 40706* A sequence of function values, approximates the corresponding limit function value, all but finitely many times. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑚𝜑    &   𝑚𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝑋𝐷)    &   (𝜑𝑌 ∈ ℝ+)       (𝜑 → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)(((𝐹𝑚)‘𝑋) ∈ ℝ ∧ (abs‘(((𝐹𝑚)‘𝑋) − (𝐺𝑋))) < 𝑌))

Theoremclimfveqf 40707* Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   𝑘𝐹    &   𝑘𝐺    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺))

Theoremclimmptf 40708* Exhibit a function 𝐺 with the same convergence properties as the not-quite-function 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   𝑍 = (ℤ𝑀)    &   𝐺 = (𝑘𝑍 ↦ (𝐹𝑘))       (𝜑 → (𝐹𝐴𝐺𝐴))

Theoremclimfveqmpt3 40709* Two functions that are eventually equal to one another have the same limit. TODO: this is more general than climfveqmpt 40698 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑𝑍𝐴)    &   (𝜑𝑍𝐶)    &   ((𝜑𝑘𝑍) → 𝐵𝑈)    &   ((𝜑𝑘𝑍) → 𝐵 = 𝐷)       (𝜑 → ( ⇝ ‘(𝑘𝐴𝐵)) = ( ⇝ ‘(𝑘𝐶𝐷)))

Theoremclimeldmeqf 40710* Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   𝑘𝐹    &   𝑘𝐺    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ))

Theoremclimreclmpt 40711* The limit of B convergent real sequence is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   (𝜑 → (𝑘𝑍𝐴) ⇝ 𝐵)       (𝜑𝐵 ∈ ℝ)

Theoremlimsupref 40712* If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))       (𝜑 → (lim sup‘𝐹) ∈ ℝ)

Theoremlimsupbnd1f 40713* If a sequence is eventually at most 𝐴, then the limsup is also at most 𝐴. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐹:𝐵⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴))       (𝜑 → (lim sup‘𝐹) ≤ 𝐴)

Theoremclimbddf 40714* A converging sequence of complex numbers is bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝐹    &   𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘𝑍 (𝐹𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) ≤ 𝑥)

Theoremclimeqf 40715* Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   𝑘𝐹    &   𝑘𝐺    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (𝐹𝐴𝐺𝐴))

Theoremclimeldmeqmpt3 40716* Two functions that are eventually equal, either both are convergent or both are divergent. TODO: this is more general than climeldmeqmpt 40695 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑𝑍𝐴)    &   (𝜑𝑍𝐶)    &   ((𝜑𝑘𝑍) → 𝐵𝑈)    &   ((𝜑𝑘𝑍) → 𝐵 = 𝐷)       (𝜑 → ((𝑘𝐴𝐵) ∈ dom ⇝ ↔ (𝑘𝐶𝐷) ∈ dom ⇝ ))

Theoremlimsupcld 40717 Closure of the superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹𝑉)       (𝜑 → (lim sup‘𝐹) ∈ ℝ*)

Theoremclimfv 40718 The limit of a convergent sequence, expressed as the function value of the convergence relation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐹𝐴𝐴 = ( ⇝ ‘𝐹))

Theoremlimsupval3 40719* The superior limit of an infinite sequence 𝐹 of extended real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))       (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))

Theoremclimfveqmpt2 40720* Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝑍𝐴)    &   (𝜑𝑍𝐵)    &   ((𝜑𝑘𝑍) → 𝐶𝑈)       (𝜑 → ( ⇝ ‘(𝑘𝐴𝐶)) = ( ⇝ ‘(𝑘𝐵𝐶)))

Theoremlimsup0 40721 The superior limit of the empty set (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(lim sup‘∅) = -∞

Theoremclimeldmeqmpt2 40722* Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑊)    &   (𝜑𝐵𝑉)    &   (𝜑𝑍𝐴)    &   (𝜑𝑍𝐵)    &   ((𝜑𝑘𝑍) → 𝐶𝑈)       (𝜑 → ((𝑘𝐴𝐶) ∈ dom ⇝ ↔ (𝑘𝐵𝐶) ∈ dom ⇝ ))

Theoremlimsupresre 40723 The supremum limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹𝑉)       (𝜑 → (lim sup‘(𝐹 ↾ ℝ)) = (lim sup‘𝐹))

Theoremclimeqmpt 40724* Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑍𝐴)    &   (𝜑𝑍𝐵)    &   ((𝜑𝑥𝑍) → 𝐶𝑈)       (𝜑 → ((𝑥𝐴𝐶) ⇝ 𝐷 ↔ (𝑥𝐵𝐶) ⇝ 𝐷))

Theoremclimfvd 40725 The limit of a convergent sequence, expressed as the function value of the convergence relation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹𝐴)       (𝜑𝐴 = ( ⇝ ‘𝐹))

Theoremlimsuplesup 40726 An upper bound for the superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹𝑉)    &   (𝜑𝐾 ∈ ℝ)       (𝜑 → (lim sup‘𝐹) ≤ sup(((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*), ℝ*, < ))

Theoremlimsupresico 40727 The superior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℝ)    &   𝑍 = (𝑀[,)+∞)    &   (𝜑𝐹𝑉)       (𝜑 → (lim sup‘(𝐹𝑍)) = (lim sup‘𝐹))

Theoremlimsuppnfdlem 40728* If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))    &   𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝜑 → (lim sup‘𝐹) = +∞)

Theoremlimsuppnfd 40729* If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))       (𝜑 → (lim sup‘𝐹) = +∞)

Theoremlimsupresuz 40730 If the real part of the domain of a function is a subset of the integers, the superior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ)       (𝜑 → (lim sup‘(𝐹𝑍)) = (lim sup‘𝐹))

Theoremlimsupub 40731* If the limsup is not +∞, then the function is eventually bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝜑    &   𝑗𝐹    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   (𝜑 → (lim sup‘𝐹) ≠ +∞)       (𝜑 → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))

Theoremlimsupres 40732 The superior limit of a restriction is less than or equal to the original superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹𝑉)       (𝜑 → (lim sup‘(𝐹𝐶)) ≤ (lim sup‘𝐹))

Theoremcliminf2lem 40733* A convergent, nonincreasing sequence, converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶ℝ)    &   ((𝜑𝑘𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘))       (𝜑𝐹 ⇝ inf(ran 𝐹, ℝ*, < ))

Theoremcliminf2 40734* A convergent, nonincreasing sequence, converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶ℝ)    &   ((𝜑𝑘𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘))       (𝜑𝐹 ⇝ inf(ran 𝐹, ℝ*, < ))

Theoremlimsupvaluz 40735* The superior limit, when the domain of the function is a set of upper integers (the first condition is needed, otherwise the l.h.s. would be -∞ and the r.h.s. would be +∞). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ*, < ))

Theoremlimsupresuz2 40736 If the domain of a function is a subset of the integers, the superior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑 → dom 𝐹 ⊆ ℤ)       (𝜑 → (lim sup‘(𝐹𝑍)) = (lim sup‘𝐹))

Theoremlimsuppnflem 40737* If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)       (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))

Theoremlimsuppnf 40738* If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)       (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))

Theoremlimsupubuzlem 40739* If the limsup is not +∞, then the function is bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝜑    &   𝑗𝑋    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐾 ∈ ℝ)    &   (𝜑 → ∀𝑗𝑍 (𝐾𝑗 → (𝐹𝑗) ≤ 𝑌))    &   𝑁 = if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾))    &   𝑊 = sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹𝑗)), ℝ, < )    &   𝑋 = if(𝑊𝑌, 𝑌, 𝑊)       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)

Theoremlimsupubuz 40740* For a real-valued function on a set of upper integers, if the superior limit is not +∞, then the function is bounded above. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim sup‘𝐹) ≠ +∞)       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)

Theoremcliminf2mpt 40741* A bounded below, monotonic nonincreasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   𝑗𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)    &   (𝑘 = 𝑗𝐵 = 𝐶)    &   ((𝜑𝑘𝑍𝑗 = (𝑘 + 1)) → 𝐶𝐵)    &   (𝜑 → (𝑘𝑍𝐵) ∈ dom ⇝ )       (𝜑 → (𝑘𝑍𝐵) ⇝ inf(ran (𝑘𝑍𝐵), ℝ*, < ))

Theoremcliminfmpt 40742* A bounded below, monotonic nonincreasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   𝑗𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)    &   (𝑘 = 𝑗𝐵 = 𝐶)    &   ((𝜑𝑘𝑍𝑗 = (𝑘 + 1)) → 𝐶𝐵)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥𝐵)       (𝜑 → (𝑘𝑍𝐵) ⇝ inf(ran (𝑘𝑍𝐵), ℝ*, < ))

Theoremcliminf3 40743* A convergent, nonincreasing sequence, converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   ((𝜑𝑘𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))    &   (𝜑𝐹 ∈ dom ⇝ )       (𝜑𝐹 ⇝ inf(ran 𝐹, ℝ*, < ))

Theoremlimsupvaluzmpt 40744* The superior limit, when the domain of the function is a set of upper integers (the first condition is needed, otherwise the l.h.s. would be -∞ and the r.h.s. would be +∞). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑗𝑍) → 𝐵 ∈ ℝ*)       (𝜑 → (lim sup‘(𝑗𝑍𝐵)) = inf(ran (𝑘𝑍 ↦ sup(ran (𝑗 ∈ (ℤ𝑘) ↦ 𝐵), ℝ*, < )), ℝ*, < ))

Theoremlimsupequzmpt2 40745* Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝜑    &   𝑗𝐴    &   𝑗𝐵    &   𝐴 = (ℤ𝑀)    &   𝐵 = (ℤ𝑁)    &   (𝜑𝐾𝐴)    &   (𝜑𝐾𝐵)    &   ((𝜑𝑗 ∈ (ℤ𝐾)) → 𝐶𝑉)       (𝜑 → (lim sup‘(𝑗𝐴𝐶)) = (lim sup‘(𝑗𝐵𝐶)))

Theoremlimsupubuzmpt 40746* If the limsup is not +∞, then the function is eventually bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝜑    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑗𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (lim sup‘(𝑗𝑍𝐵)) ≠ +∞)       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝐵𝑥)

Theoremlimsupmnflem 40747* The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to an upper interval of real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)    &   𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))       (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))

Theoremlimsupmnf 40748* The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to an upper interval of real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)       (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))

Theoremlimsupequzlem 40749* Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹 Fn (ℤ𝑀))    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐺 Fn (ℤ𝑁))    &   (𝜑𝐾 ∈ ℤ)    &   ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺))

Theoremlimsupequz 40750* Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑘𝜑    &   𝑘𝐹    &   𝑘𝐺    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹 Fn (ℤ𝑀))    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐺 Fn (ℤ𝑁))    &   (𝜑𝐾 ∈ ℤ)    &   ((𝜑𝑘 ∈ (ℤ𝐾)) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺))

Theoremlimsupre2lem 40751* Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)       (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 < (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑥))))

Theoremlimsupre2 40752* Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)       (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 < (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑥))))

Theoremlimsupmnfuzlem 40753* The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to a set of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))

Theoremlimsupmnfuz 40754* The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to a set of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))

Theoremlimsupequzmptlem 40755* Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝜑    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   𝐴 = (ℤ𝑀)    &   𝐵 = (ℤ𝑁)    &   ((𝜑𝑗𝐴) → 𝐶𝑉)    &   ((𝜑𝑗𝐵) → 𝐶𝑊)    &   𝐾 = if(𝑀𝑁, 𝑁, 𝑀)       (𝜑 → (lim sup‘(𝑗𝐴𝐶)) = (lim sup‘(𝑗𝐵𝐶)))

Theoremlimsupequzmpt 40756* Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝜑    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   𝐴 = (ℤ𝑀)    &   𝐵 = (ℤ𝑁)    &   ((𝜑𝑗𝐴) → 𝐶𝑉)    &   ((𝜑𝑗𝐵) → 𝐶𝑊)       (𝜑 → (lim sup‘(𝑗𝐴𝐶)) = (lim sup‘(𝑗𝐵𝐶)))

Theoremlimsupre2mpt 40757* Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)       (𝜑 → ((lim sup‘(𝑥𝐴𝐵)) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥𝐴 (𝑘𝑥𝑦 < 𝐵) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑥𝐴 (𝑘𝑥𝐵 < 𝑦))))

Theoremlimsupequzmptf 40758* Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝜑    &   𝑗𝐴    &   𝑗𝐵    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   𝐴 = (ℤ𝑀)    &   𝐵 = (ℤ𝑁)    &   ((𝜑𝑗𝐴) → 𝐶𝑉)    &   ((𝜑𝑗𝐵) → 𝐶𝑊)       (𝜑 → (lim sup‘(𝑗𝐴𝐶)) = (lim sup‘(𝑗𝐵𝐶)))

Theoremlimsupre3lem 40759* Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)       (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))))

Theoremlimsupre3 40760* Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ*)       (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))))

Theoremlimsupre3mpt 40761* Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)       (𝜑 → ((lim sup‘(𝑥𝐴𝐵)) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥𝐴 (𝑘𝑥𝑦𝐵) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑥𝐴 (𝑘𝑥𝐵𝑦))))

Theoremlimsupre3uzlem 40762* Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, infinitely often; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))

Theoremlimsupre3uz 40763* Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, infinitely often; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))

Theoremlimsupreuz 40764* Given a function on the reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, infinitely often; 2. there is a real number that is greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)))

Theoremlimsupvaluz2 40765* The superior limit, when the domain of a real-valued function is a set of upper integers, and the superior limit is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim sup‘𝐹) ∈ ℝ)       (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑘)), ℝ*, < )), ℝ, < ))

Theoremlimsupreuzmpt 40766* Given a function on the reals, defined on a set of upper integers, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, infinitely often; 2. there is a real number that is greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑗𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑗𝑍) → 𝐵 ∈ ℝ)       (𝜑 → ((lim sup‘(𝑗𝑍𝐵)) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥𝐵 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝐵𝑥)))

Theoremsupcnvlimsup 40767* 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 40768* 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 40769 If (/) is a complex number, then it converges to itself. (see 0ncn 10270 and its comment ; see also the comment in climlimsupcex 40796) (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(∅ ∈ ℂ → ∅ ⇝ ∅)

Theoremclimuzlem 40770* 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 40771* 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 40772* 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 40773* 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 40774* 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 40775* 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 40776* 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 40777* 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)

20.33.7.1  Inferior limit (lim inf)

Syntaxclsi 40778 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 40779* 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 40780* 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 40781 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 40782* 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 40783* 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 40784 Closure of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐹𝑉       (lim sup‘𝐹) ∈ ℝ*

Theoremliminfgf 40785 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 40786* 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 40787 A sequence of real numbers converges if and only if it converges to its superior limit. The first hypothesis is needed (see climlimsupcex 40796 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 40788* 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 40789* 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 40790 Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐹𝑉 → (lim inf‘𝐹) ∈ ℝ*)

Theoremliminfvald 40791* 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 40792* 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 40793 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 40794 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 40795* 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 40796 Counterexample for climlimsup 40787, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 10270 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 40797 Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)       (𝜑 → (lim inf‘𝐹) ∈ ℝ*)

Theoremliminfresico 40798 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 40799* 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 40800 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

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