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Theorem List for Metamath Proof Explorer - 40201-40300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlimclner 40201 For a limit point, both from the left and from the right, of the domain, the limit of the function exits only if the left and the right limits are equal. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐽 = (topGen‘ran (,))    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))))    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞))))    &   (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) lim 𝐵))    &   (𝜑𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) lim 𝐵))    &   (𝜑𝐿𝑅)       (𝜑 → (𝐹 lim 𝐵) = ∅)

Theoremsublimc 40202* Subtraction of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ (𝐵𝐶))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)    &   (𝜑𝐸 ∈ (𝐹 lim 𝐷))    &   (𝜑𝐼 ∈ (𝐺 lim 𝐷))       (𝜑 → (𝐸𝐼) ∈ (𝐻 lim 𝐷))

Theoremreclimc 40203* Limit of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴 ↦ (1 / 𝐵))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (ℂ ∖ {0}))    &   (𝜑𝐶 ∈ (𝐹 lim 𝐷))    &   (𝜑𝐶 ≠ 0)       (𝜑 → (1 / 𝐶) ∈ (𝐺 lim 𝐷))

Theoremclim0cf 40204* Express the predicate 𝐹 converges to 0. Similar to clim 14269, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘𝐵) < 𝑥))

Theoremlimclr 40205 For a limit point, both from the left and from the right, of the domain, the limit of the function exits only if the left and the right limits are equal. In this case, the three limits coincide. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐽 = (topGen‘ran (,))    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))))    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞))))    &   (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) lim 𝐵))    &   (𝜑𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) lim 𝐵))       (𝜑 → (((𝐹 lim 𝐵) ≠ ∅ ↔ 𝐿 = 𝑅) ∧ (𝐿 = 𝑅𝐿 ∈ (𝐹 lim 𝐵))))

Theoremdivlimc 40206* Limit of the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ (𝐵 / 𝐶))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ (ℂ ∖ {0}))    &   (𝜑𝑋 ∈ (𝐹 lim 𝐷))    &   (𝜑𝑌 ∈ (𝐺 lim 𝐷))    &   (𝜑𝑌 ≠ 0)    &   ((𝜑𝑥𝐴) → 𝐶 ≠ 0)       (𝜑 → (𝑋 / 𝑌) ∈ (𝐻 lim 𝐷))

Theoremexpfac 40207* Factorial grows faster than exponential. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → 𝐹 ⇝ 0)

Theoremclimconstmpt 40208* A constant sequence converges to its value. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴 ∈ ℂ)       (𝜑 → (𝑥𝑍𝐴) ⇝ 𝐴)

Theoremclimresmpt 40209* A function restricted to upper integers converges iff the original function converges. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑍 = (ℤ𝑀)    &   𝐹 = (𝑥𝑍𝐴)    &   (𝜑𝑁𝑍)    &   𝐺 = (𝑥 ∈ (ℤ𝑁) ↦ 𝐴)       (𝜑 → (𝐺𝐵𝐹𝐵))

Theoremclimsubmpt 40210* Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)    &   (𝜑 → (𝑘𝑍𝐴) ⇝ 𝐶)    &   (𝜑 → (𝑘𝑍𝐵) ⇝ 𝐷)       (𝜑 → (𝑘𝑍 ↦ (𝐴𝐵)) ⇝ (𝐶𝐷))

Theoremclimsubc2mpt 40211* Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → (𝑘𝑍𝐴) ⇝ 𝐶)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝑘𝑍 ↦ (𝐴𝐵)) ⇝ (𝐶𝐵))

Theoremclimsubc1mpt 40212* Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)    &   (𝜑 → (𝑘𝑍𝐵) ⇝ 𝐶)       (𝜑 → (𝑘𝑍 ↦ (𝐴𝐵)) ⇝ (𝐴𝐶))

Theoremfnlimfv 40213* The value of the limit function 𝐺 at any point of its domain 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐷    &   𝑥𝐹    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝑋𝐷)       (𝜑 → (𝐺𝑋) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))

Theoremclimreclf 40214* The limit of a convergent real sequence is real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)       (𝜑𝐴 ∈ ℝ)

Theoremclimeldmeq 40215* Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ))

Theoremclimf2 40216* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. Similar to clim 14269, but without the disjoint var constraint 𝜑𝑘 and 𝐹𝑘. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))

Theoremfnlimcnv 40217* The sequence of function values converges to the value of the limit function 𝐺 at any point of its domain 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝑋𝐷)       (𝜑 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋)) ⇝ (𝐺𝑋))

Theoremclimeldmeqmpt 40218* Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑅)    &   (𝜑𝑍𝐴)    &   ((𝜑𝑘𝐴) → 𝐵𝑉)    &   (𝜑𝐶𝑆)    &   (𝜑𝑍𝐶)    &   ((𝜑𝑘𝐶) → 𝐷𝑊)    &   ((𝜑𝑘𝑍) → 𝐵 = 𝐷)       (𝜑 → ((𝑘𝐴𝐵) ∈ dom ⇝ ↔ (𝑘𝐶𝐷) ∈ dom ⇝ ))

Theoremclimfveq 40219* Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺))

Theoremclim2f2 40220* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 14269. Similar to clim2 14279, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))

Theoremclimfveqmpt 40221* Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑅)    &   (𝜑𝑍𝐴)    &   ((𝜑𝑘𝐴) → 𝐵𝑉)    &   (𝜑𝐶𝑆)    &   (𝜑𝑍𝐶)    &   ((𝜑𝑘𝐶) → 𝐷𝑊)    &   ((𝜑𝑘𝑍) → 𝐵 = 𝐷)       (𝜑 → ( ⇝ ‘(𝑘𝐴𝐵)) = ( ⇝ ‘(𝑘𝐶𝐷)))

Theoremclimd 40222* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋))

Theoremclim2d 40223* The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋))

Theoremfnlimfvre 40224* 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 40225* 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 40226* 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 40227* 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 40228* 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 40229* 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 40230* 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 40231* 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 40232* Two functions that are eventually equal to one another have the same limit. TODO: this is more general than climfveqmpt 40221 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 40233* 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 40234* 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 40235* 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 40236* 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 40237* 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 40238* 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 40239* Two functions that are eventually equal, either both are convergent or both are divergent. TODO: this is more general than climeldmeqmpt 40218 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 40240 Closure of the superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹𝑉)       (𝜑 → (lim sup‘𝐹) ∈ ℝ*)

Theoremclimfv 40241 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 40242* 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 40243* 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 40244 The superior limit of the empty set (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(lim sup‘∅) = -∞

Theoremclimeldmeqmpt2 40245* 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 40246 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 40247* 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 40248 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 40249 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 40250 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 40251* 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 40252* 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 40253 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 40254* 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 40255 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 40256* A convergent, non-increasing 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 40257* A convergent, non-increasing 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 40258* 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 40259 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 40260* 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 40261* 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 40262* 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 40263* 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 40264* A bounded below, monotonic non increasing 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 40265* A bounded below, monotonic non increasing 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 40266* A convergent, non-increasing 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 40267* 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 40268* 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 40269* 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 40270* 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 40271* 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 40272* 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 40273* 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 40274* 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 40275* 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 40276* 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 40277* 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 40278* 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 40279* 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 40280* 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 40281* 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 40282* 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 or equal than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger or equal 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))))

Theoremlimsupre3 40283* 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 or equal than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger or equal 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))))

Theoremlimsupre3mpt 40284* 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 or equal than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger or equal 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 (𝑘𝑥𝑦𝐵) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑥𝐴 (𝑘𝑥𝐵𝑦))))

Theoremlimsupre3uzlem 40285* 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 or equal than the function, infinitely often; 2. there is a real number that is eventually larger or equal than 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 40286* 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 or equal than the function, infinitely often; 2. there is a real number that is eventually larger or equal than 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 40287* 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 smaller or equal than the function, infinitely often; 2. there is a real number that is larger or equal than 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 40288* 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 40289* 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 smaller or equal than the function, infinitely often; 2. there is a real number that is larger or equal than 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 40290* 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 40291* 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 40292 If (/) is a complex number, then it converges to itself. (see 0ncn 9992 and its comment ; see also the comment in climlimsupcex 40319) (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(∅ ∈ ℂ → ∅ ⇝ ∅)

Theoremclimuzlem 40293* 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 40294* 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 40295* 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 40296* 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 40297* 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 40298* 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 40299* 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 40300* 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)

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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