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Theorem List for Metamath Proof Explorer - 42401-42500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlimsupresxr 42401 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 42402 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 42403* 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 42404 Counterexample for climlimsup 42395, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 10548 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 42405 Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)       (𝜑 → (lim inf‘𝐹) ∈ ℝ*)

Theoremliminfresico 42406 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 42407* 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 42408 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 42409 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 42410* 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 42411* The superior limit is greater than or equal to the inferior limit. The second hypothesis is needed (see liminflelimsupcex 42432 for a counterexample). The inequality can be strict, see liminfltlimsupex 42416. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)       (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))

Theoremlimsupgtlem 42412* 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 42413* 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 42414 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 42415* 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 42416 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 42417* 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 42418* 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 42419 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 42420 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 42421* 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 42422 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 42423 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 42424* 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 42425* 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 42426* 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 42427* 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 42428 The inferior limit of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(lim inf‘∅) = +∞

Theoremlimsupval4 42429* 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 42430* 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 42431* 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 42432 A counterexample for liminflelimsup 42411, showing that the second hypothesis is needed. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(lim sup‘∅) < (lim inf‘∅)

Theoremlimsupvaluz3 42433* 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 42434* 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 42435* 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 42436 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 42437* 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 42438* 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 42439* 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 42440* 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 42441 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 42442 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 42420). (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 42443 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 42423). (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 42444 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 42423). (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 42445 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 42446 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 42447* 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 42448* 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 42449* 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 42450* 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 42451* 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 42452* 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)

20.37.7.2  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 42483.

Syntaxclsxlim 42453 Extend class notation with convergence relation for limits in the extended real numbers.
class ~~>*

Definitiondf-xlim 42454 Define the convergence relation for extended real sequences. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
~~>* = (⇝𝑡‘(ordTop‘ ≤ ))

Theoremxlimrel 42455 The limit on extended reals is a relation. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Rel ~~>*

Theoremxlimres 42456 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 42457 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 42458* Restricted existential elimination rule of natural deduction. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑 → ∃𝑥𝐴 𝜓)    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)       (𝜑𝜒)

Theoremxlimclim 42459 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 42248). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐹~~>*𝐴𝐹𝐴))

Theoremxlimconst 42460* 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 42461 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 42462* 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 42463 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 42464* Lemma for cnrefiisp 42465 (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 42465* 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 42466* 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 42467* Lemma for xlimmnfv 42469: 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 42468* Lemma for xlimmnf 42476: 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 42469* 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 42470* 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 42471* Lemma for xlimpnfv 42473: 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 42472* Lemma for xlimpnfv 42473: 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 42473* 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 42474* Lemma for xlimclim2 42475. 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))

Theoremxlimclim2 42475 Given a sequence of extended reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals (see climreeq 42248), if and only if it converges to 𝐴 w.r.t. to the standard topology on the extended reals. In order for the first part of the statement to even make sense, the sequence will of course eventually become (and stay) real: showing this, is the key step of the proof. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐹~~>*𝐴𝐹𝐴))

Theoremxlimmnf 42476* 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.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑥))

Theoremxlimpnf 42477* 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.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘)))

Theoremxlimmnfmpt 42478* 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.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ*)    &   𝐹 = (𝑘𝑍𝐵)       (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝐵𝑥))

Theoremxlimpnfmpt 42479* 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.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ*)    &   𝐹 = (𝑘𝑍𝐵)       (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥𝐵))

Theoremclimxlim2lem 42480 In this lemma for climxlim2 42481 there is the additional assumption that the converging function is complex-valued on the whole domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹:𝑍⟶ℂ)    &   (𝜑𝐹𝐴)       (𝜑𝐹~~>*𝐴)

Theoremclimxlim2 42481 A sequence of extended reals, converging w.r.t. the standard topology on the complex numbers is a converging sequence w.r.t. the standard topology on the extended reals. This is non-trivial, because +∞ and -∞ could, in principle, be complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹𝐴)       (𝜑𝐹~~>*𝐴)

Theoremdfxlim2v 42482* An alternative definition for the convergence relation in the extended real numbers. This resembles what's found in most textbooks: three distinct definitions for the same symbol (limit of a sequence). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘)))))

Theoremdfxlim2 42483* An alternative definition for the convergence relation in the extended real numbers. This resembles what's found in most textbooks: three distinct definitions for the same symbol (limit of a sequence). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘)))))

Theoremclimresd 42484 A function restricted to upper integers converges iff the original function converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)       (𝜑 → ((𝐹 ↾ (ℤ𝑀)) ⇝ 𝐴𝐹𝐴))

Theoremclimresdm 42485 A real function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ (𝐹 ↾ (ℤ𝑀)) ∈ dom ⇝ ))

Theoremdmclimxlim 42486 A real valued sequence that converges w.r.t. the topology on the complex numbers, converges w.r.t. the topology on the extended reals (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑𝐹 ∈ dom ⇝ )       (𝜑𝐹 ∈ dom ~~>*)

Theoremxlimmnflimsup2 42487 A sequence of extended reals converges to -∞ if and only if its superior limit is also -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞))

Theoremxlimuni 42488 An infinite sequence converges to at most one limit (w.r.t. to the standard topology on the extended reals). (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝐹~~>*𝐴)    &   (𝜑𝐹~~>*𝐵)       (𝜑𝐴 = 𝐵)

Theoremxlimclimdm 42489 A sequence of extended reals that converges to a real w.r.t. the standard topology on the extended reals, also converges w.r.t. to the standard topology on the complex numbers. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹~~>*𝐴)    &   (𝜑𝐴 ∈ ℝ)       (𝜑𝐹 ∈ dom ⇝ )

Theoremxlimfun 42490 The convergence relation on the extended reals is a function. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
Fun ~~>*

Theoremxlimmnflimsup 42491 If a sequence of extended reals converges to -∞ then its superior limit is also -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹~~>*-∞)       (𝜑 → (lim sup‘𝐹) = -∞)

Theoremxlimdm 42492 Two ways to express that a function has a limit. (The expression (~~>*‘𝐹) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*‘𝐹))

Theoremxlimpnfxnegmnf2 42493* A sequence converges to +∞ if and only if its negation converges to -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*+∞ ↔ (𝑗𝑍 ↦ -𝑒(𝐹𝑗))~~>*-∞))

Theoremxlimresdm 42494 A function converges in the extended reals iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝐹 ∈ (ℝ*pm ℂ))    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐹 ∈ dom ~~>* ↔ (𝐹 ↾ (ℤ𝑀)) ∈ dom ~~>*))

Theoremxlimpnfliminf 42495 If a sequence of extended reals converges to +∞ then its superior limit is also +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹~~>*+∞)       (𝜑 → (lim inf‘𝐹) = +∞)

Theoremxlimpnfliminf2 42496 A sequence of extended reals converges to +∞ if and only if its superior limit is also +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))

Theoremxlimliminflimsup 42497 A sequence of extended reals converges if and only if its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))

Theoremxlimlimsupleliminf 42498 A sequence of extended reals converges if and only if its superior limit is smaller than or equal to its inferior limit. (Contributed by Glauco Siliprandi, 2-Dec-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))

20.37.8  Trigonometry

Theoremcoseq0 42499 A complex number whose cosine is zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℂ → ((cos‘𝐴) = 0 ↔ ((𝐴 / π) + (1 / 2)) ∈ ℤ))

Theoremsinmulcos 42500 Multiplication formula for sine and cosine. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (cos‘𝐵)) = (((sin‘(𝐴 + 𝐵)) + (sin‘(𝐴𝐵))) / 2))

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