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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | supcnvlimsup 41901* | If a function on a set of upper integers has a real superior limit, the supremum of the rightmost parts of the function, converges to that superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )) ⇝ (lim sup‘𝐹)) | ||
Theorem | supcnvlimsupmpt 41902* | If a function on a set of upper integers has a real superior limit, the supremum of the rightmost parts of the function, converges to that superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝑘) ↦ 𝐵), ℝ*, < )) ⇝ (lim sup‘(𝑗 ∈ 𝑍 ↦ 𝐵))) | ||
Theorem | 0cnv 41903 | If (/) is a complex number, then it converges to itself. (see 0ncn 10544 and its comment ; see also the comment in climlimsupcex 41930) (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) | ||
Theorem | climuzlem 41904* | Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))) | ||
Theorem | climuz 41905* | Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))) | ||
Theorem | lmbr3v 41906* | Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) | ||
Theorem | climisp 41907* | If a sequence converges to an isolated point (w.r.t. the standard topology on the complex numbers) then the sequence eventually becomes that point. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝐹‘𝑘) ≠ 𝐴) → 𝑋 ≤ (abs‘((𝐹‘𝑘) − 𝐴))) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) = 𝐴) | ||
Theorem | lmbr3 41908* | Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) | ||
Theorem | climrescn 41909* | A sequence converging w.r.t. the standard topology on the complex numbers, eventually becomes a sequence of complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 Fn 𝑍) & ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) | ||
Theorem | climxrrelem 41910* | If a seqence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐷 ≤ (abs‘(+∞ − 𝐴))) & ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐷 ≤ (abs‘(-∞ − 𝐴))) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) | ||
Theorem | climxrre 41911* | If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued (the weaker hypothesis 𝐹 ∈ dom ⇝ is probably not enough, since in principle we could have +∞ ∈ ℂ and -∞ ∈ ℂ). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) | ||
Syntax | clsi 41912 | Extend class notation to include the liminf function. (actually, it makes sense for any extended real function defined on a subset of RR which is not upper-bounded) |
class lim inf | ||
Definition | df-liminf 41913* | Define the inferior limit of a sequence of extended real numbers. (Contributed by GS, 2-Jan-2022.) |
⊢ lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) | ||
Theorem | limsuplt2 41914* | The defining property of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐵 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑘 ∈ ℝ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴)) | ||
Theorem | liminfgord 41915 | Ordering property of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → inf(((𝐹 “ (𝐴[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝐵[,)+∞)) ∩ ℝ*), ℝ*, < )) | ||
Theorem | limsupvald 41916* | The superior limit of a sequence 𝐹 of extended real numbers is the infimum of the set of suprema of all restrictions of 𝐹 to an upperset of reals . (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < )) | ||
Theorem | limsupresicompt 41917* | The superior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ 𝑍 = (𝑀[,)+∞) ⇒ ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵))) | ||
Theorem | limsupcli 41918 | Closure of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐹 ∈ 𝑉 ⇒ ⊢ (lim sup‘𝐹) ∈ ℝ* | ||
Theorem | liminfgf 41919 | Closure of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⇒ ⊢ 𝐺:ℝ⟶ℝ* | ||
Theorem | liminfval 41920* | The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⇒ ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < )) | ||
Theorem | climlimsup 41921 | A sequence of real numbers converges if and only if it converges to its superior limit. The first hypothesis is needed (see climlimsupcex 41930 for a counterexample) (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim sup‘𝐹))) | ||
Theorem | limsupge 41922* | The defining property of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐵 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑘 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) | ||
Theorem | liminfgval 41923* | Value of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⇒ ⊢ (𝑀 ∈ ℝ → (𝐺‘𝑀) = inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < )) | ||
Theorem | liminfcl 41924 | Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) ∈ ℝ*) | ||
Theorem | liminfvald 41925* | The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < )) | ||
Theorem | liminfval5 41926* | The inferior limit of an infinite sequence 𝐹 of extended real numbers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < )) | ||
Theorem | limsupresxr 41927 | The superior limit of a function only depends on the restriction of that function to the preimage of the set of extended reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝐹) & ⊢ 𝐴 = (◡𝐹 “ ℝ*) ⇒ ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝐴)) = (lim sup‘𝐹)) | ||
Theorem | liminfresxr 41928 | The inferior limit of a function only depends on the preimage of the extended real part. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝐹) & ⊢ 𝐴 = (◡𝐹 “ ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝐴)) = (lim inf‘𝐹)) | ||
Theorem | liminfval2 41929* | The superior limit, relativized to an unbounded set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = sup((𝐺 “ 𝐴), ℝ*, < )) | ||
Theorem | climlimsupcex 41930 | Counterexample for climlimsup 41921, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 10544 and its comment) (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ ¬ 𝑀 ∈ ℤ & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐹 = ∅ ⇒ ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹:𝑍⟶ℝ ∧ 𝐹 ∈ dom ⇝ ∧ ¬ 𝐹 ⇝ (lim sup‘𝐹))) | ||
Theorem | liminfcld 41931 | Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ*) | ||
Theorem | liminfresico 41932 | The inferior limit doesn't change when a function is restricted to an upperset of reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ 𝑍 = (𝑀[,)+∞) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹)) | ||
Theorem | limsup10exlem 41933* | The range of the given function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) & ⊢ (𝜑 → 𝐾 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐹 “ (𝐾[,)+∞)) = {0, 1}) | ||
Theorem | limsup10ex 41934 | The superior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) ⇒ ⊢ (lim sup‘𝐹) = 1 | ||
Theorem | liminf10ex 41935 | The inferior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) ⇒ ⊢ (lim inf‘𝐹) = 0 | ||
Theorem | liminflelimsuplem 41936* | The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) | ||
Theorem | liminflelimsup 41937* | The superior limit is greater than or equal to the inferior limit. The second hypothesis is needed (see liminflelimsupcex 41958 for a counterexample). The inequality can be strict, see liminfltlimsupex 41942. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) | ||
Theorem | limsupgtlem 41938* | For any positive real, the superior limit of F is larger than any of its values at large enough arguments, up to that positive real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) | ||
Theorem | limsupgt 41939* | Given a sequence of real numbers, there exists an upper part of the sequence that's appxoximated from below by the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) | ||
Theorem | liminfresre 41940 | The inferior limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ ℝ)) = (lim inf‘𝐹)) | ||
Theorem | liminfresicompt 41941* | The inferior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ 𝑍 = (𝑀[,)+∞) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵)) = (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵))) | ||
Theorem | liminfltlimsupex 41942 | An example where the lim inf is strictly smaller than the lim sup. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) ⇒ ⊢ (lim inf‘𝐹) < (lim sup‘𝐹) | ||
Theorem | liminfgelimsup 41943* | The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹))) | ||
Theorem | liminfvalxr 41944* | Alternate definition of lim inf when 𝐹 is an extended real-valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒(𝐹‘𝑥)))) | ||
Theorem | liminfresuz 41945 | If the real part of the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹)) | ||
Theorem | liminflelimsupuz 41946 | The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) | ||
Theorem | liminfvalxrmpt 41947* | Alternate definition of lim inf when 𝐹 is an extended real-valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) | ||
Theorem | liminfresuz2 41948 | If the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → dom 𝐹 ⊆ ℤ) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹)) | ||
Theorem | liminfgelimsupuz 41949 | The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹))) | ||
Theorem | liminfval4 41950* | Alternate definition of lim inf when the given function is eventually real-valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝐵))) | ||
Theorem | liminfval3 41951* | Alternate definition of lim inf when the given function is eventually extended real-valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) | ||
Theorem | liminfequzmpt2 41952* | Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑗𝐴 & ⊢ Ⅎ𝑗𝐵 & ⊢ 𝐴 = (ℤ≥‘𝑀) & ⊢ 𝐵 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐾 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim inf‘(𝑗 ∈ 𝐵 ↦ 𝐶))) | ||
Theorem | liminfvaluz 41953* | Alternate definition of lim inf for an extended real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝑒𝐵))) | ||
Theorem | liminf0 41954 | The inferior limit of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (lim inf‘∅) = +∞ | ||
Theorem | limsupval4 41955* | Alternate definition of lim inf when the given a function is eventually extended real-valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim inf‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) | ||
Theorem | liminfvaluz2 41956* | Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝐵))) | ||
Theorem | liminfvaluz3 41957* | Alternate definition of lim inf for an extended real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑘)))) | ||
Theorem | liminflelimsupcex 41958 | A counterexample for liminflelimsup 41937, showing that the second hypothesis is needed. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (lim sup‘∅) < (lim inf‘∅) | ||
Theorem | limsupvaluz3 41959* | Alternate definition of lim inf for an extended real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ -𝑒𝐵))) | ||
Theorem | liminfvaluz4 41960* | Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)))) | ||
Theorem | limsupvaluz4 41961* | Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ -𝐵))) | ||
Theorem | climliminflimsupd 41962 | If a sequence of real numbers converges, its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹)) | ||
Theorem | liminfreuzlem 41963* | Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))) | ||
Theorem | liminfreuz 41964* | Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))) | ||
Theorem | liminfltlem 41965* | Given a sequence of real numbers, there exists an upper part of the sequence that's approximated from above by the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋)) | ||
Theorem | liminflt 41966* | Given a sequence of real numbers, there exists an upper part of the sequence that's approximated from above by the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋)) | ||
Theorem | climliminf 41967 | A sequence of real numbers converges if and only if it converges to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim inf‘𝐹))) | ||
Theorem | liminflimsupclim 41968 | A sequence of real numbers converges if its inferior limit is real, and it is greater than or equal to the superior limit (in such a case, they are actually equal, see liminflelimsupuz 41946). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ) & ⊢ (𝜑 → (lim sup‘𝐹) ≤ (lim inf‘𝐹)) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | ||
Theorem | climliminflimsup 41969 | A sequence of real numbers converges if and only if its inferior limit is real and it is greater than or equal to its superior limit (in such a case, they are actually equal, see liminfgelimsupuz 41949). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))) | ||
Theorem | climliminflimsup2 41970 | A sequence of real numbers converges if and only if its superior limit is real and it is less than or equal to its inferior limit (in such a case, they are actually equal, see liminfgelimsupuz 41949). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))) | ||
Theorem | climliminflimsup3 41971 | A sequence of real numbers converges if and only if its inferior limit is real and equal to its superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim inf‘𝐹) = (lim sup‘𝐹)))) | ||
Theorem | climliminflimsup4 41972 | A sequence of real numbers converges if and only if its superior limit is real and equal to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim inf‘𝐹) = (lim sup‘𝐹)))) | ||
Theorem | limsupub2 41973* | A extended real valued function, with limsup that is not +∞, is eventually less than +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < +∞)) | ||
Theorem | limsupubuz2 41974* | A sequence with values in the extended reals, and with limsup that is not +∞, is eventually less than +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) < +∞) | ||
Theorem | xlimpnfxnegmnf 41975* | A sequence converges to +∞ if and only if its negation converges to -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
⊢ Ⅎ𝑗𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) | ||
Theorem | liminflbuz2 41976* | A sequence with values in the extended reals, and with liminf that is not -∞, is eventually greater than -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → (lim inf‘𝐹) ≠ -∞) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-∞ < (𝐹‘𝑗)) | ||
Theorem | liminfpnfuz 41977* | The inferior limit of a function is +∞ if and only if every real number is the lower bound of the restriction of the function to a set of upper integers. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim inf‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) | ||
Theorem | liminflimsupxrre 41978* | A sequence with values in the extended reals, and with real liminf and limsup, is eventually real. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞) & ⊢ (𝜑 → (lim inf‘𝐹) ≠ -∞) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶ℝ) | ||
Textbooks generally use a single symbol to denote the limit of a sequence of real numbers. But then, three distinct definitions are usually given: one for the case of convergence to a real number, one for the case of limit to +∞ and one for the case of limit to -∞. It turns out that these three definitions can be expressed as the limit w.r.t. to the standard topology on the extended reals. In this section, a relation ~~>* is defined that captures all three definitions (and can be applied to sequences of extended reals, also), see dfxlim2 42009. | ||
Syntax | clsxlim 41979 | Extend class notation with convergence relation for limits in the extended real numbers. |
class ~~>* | ||
Definition | df-xlim 41980 | Define the convergence relation for extended real sequences. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | ||
Theorem | xlimrel 41981 | The limit on extended reals is a relation. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ Rel ~~>* | ||
Theorem | xlimres 41982 | A function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑀))~~>*𝐴)) | ||
Theorem | xlimcl 41983 | The limit of a sequence of extended real numbers is an extended real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝐹~~>*𝐴 → 𝐴 ∈ ℝ*) | ||
Theorem | rexlimddv2 41984* | Restricted existential elimination rule of natural deduction. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) & ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | xlimclim 41985 | Given a sequence of reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals, if and only if it converges to 𝐴 w.r.t. to the standard topology on the extended reals (see climreeq 41774). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) | ||
Theorem | xlimconst 41986* | A constant sequence converges to its value, w.r.t. the standard topology on the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 Fn 𝑍) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) ⇒ ⊢ (𝜑 → 𝐹~~>*𝐴) | ||
Theorem | climxlim 41987 | A converging sequence in the reals is a converging sequence in the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) ⇒ ⊢ (𝜑 → 𝐹~~>*𝐴) | ||
Theorem | xlimbr 41988* | Express the binary relation "sequence 𝐹 converges to point 𝑃 " w.r.t. the standard topology on the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ 𝐽 = (ordTop‘ ≤ ) ⇒ ⊢ (𝜑 → (𝐹~~>*𝑃 ↔ (𝑃 ∈ ℝ* ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) | ||
Theorem | fuzxrpmcn 41989 | A function mapping from an upper set of integers to the extended reals is a partial map on the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) | ||
Theorem | cnrefiisplem 41990* | Lemma for cnrefiisp 41991 (some local definitions are used). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ 𝐶 = (ℝ ∪ 𝐵) & ⊢ 𝐷 = ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}) & ⊢ 𝑋 = inf(𝐷, ℝ*, < ) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) | ||
Theorem | cnrefiisp 41991* | A non-real, complex number is an isolated point w.r.t. the union of the reals with any finite set (the extended reals is an example of such a union). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ 𝐶 = (ℝ ∪ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) | ||
Theorem | xlimxrre 41992* | If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued (the weaker hypothesis 𝐹 ∈ dom ⇝ is probably not enough, since in principle we could have +∞ ∈ ℂ and -∞ ∈ ℂ). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐹~~>*𝐴) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) | ||
Theorem | xlimmnfvlem1 41993* | Lemma for xlimmnfv 41995: the "only if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐹~~>*-∞) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑋) | ||
Theorem | xlimmnfvlem2 41994* | Lemma for xlimmnf 42002: the "if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) < 𝑥) ⇒ ⊢ (𝜑 → 𝐹~~>*-∞) | ||
Theorem | xlimmnfv 41995* | A function converges to minus infinity if it eventually becomes (and stays) smaller than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) | ||
Theorem | xlimconst2 41996* | A sequence that eventually becomes constant, converges to its constant value (w.r.t. the standard topology on the extended reals). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) = 𝐴) ⇒ ⊢ (𝜑 → 𝐹~~>*𝐴) | ||
Theorem | xlimpnfvlem1 41997* | Lemma for xlimpnfv 41999: the "only if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐹~~>*+∞) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑋 ≤ (𝐹‘𝑘)) | ||
Theorem | xlimpnfvlem2 41998* | Lemma for xlimpnfv 41999: the "if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 𝐹~~>*+∞) | ||
Theorem | xlimpnfv 41999* | A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) | ||
Theorem | xlimclim2lem 42000* | Lemma for xlimclim2 42001. Here it is additionally assumed that the sequence will eventually become (and stay) real. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
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