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Theorem List for Metamath Proof Explorer - 13901-14000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremabsdivd 13901 Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))

Theoremabstrid 13902 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))

Theoremabs2difd 13903 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴𝐵)))

Theoremabs2dif2d 13904 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))

Theoremabs2difabsd 13905 Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴𝐵)))

Theoremabs3difd 13906 Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (abs‘(𝐴𝐵)) ≤ ((abs‘(𝐴𝐶)) + (abs‘(𝐶𝐵))))

Theoremabs3lemd 13907 Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑 → (abs‘(𝐴𝐶)) < (𝐷 / 2))    &   (𝜑 → (abs‘(𝐶𝐵)) < (𝐷 / 2))       (𝜑 → (abs‘(𝐴𝐵)) < 𝐷)

5.10  Elementary limits and convergence

5.10.1  Superior limit (lim sup)

Syntaxclsp 13908 Extend class notation to include the limsup function.
class lim sup

Syntaxclspold 13909 Extend class notation to include the limsup function (old version).
class lim sup

Definitiondf-limsup 13910* Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 13915 for its value. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 11-Sep-2020.)
lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))

Definitiondf-limsupOLD 13911* Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupvalOLD 13916 for its value. (Contributed by NM, 26-Oct-2005.) Obsolete version of df-limsup 13910 as of 11-Sep-2020. (New usage is discouraged.)
lim sup = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))

Theoremlimsupgord 13912 Ordering property of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → sup(((𝐹 “ (𝐵[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝐴[,)+∞)) ∩ ℝ*), ℝ*, < ))

Theoremlimsupcl 13913 Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 12-Sep-2020.)
(𝐹𝑉 → (lim sup‘𝐹) ∈ ℝ*)

TheoremlimsupclOLD 13914 Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 7-May-2016.) Obsolete version of limsupcl 13913 as of 12-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐹𝑉 → (lim sup‘𝐹) ∈ ℝ*)

Theoremlimsupval 13915* The superior limit of an infinite sequence 𝐹 of extended real numbers, which is the infimum of the set of suprema of all upper infinite subsequences of 𝐹. Definition 12-4.1 of [Gleason] p. 175. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 12-Sep-2014.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝐹𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))

TheoremlimsupvalOLD 13916* The superior limit of an infinite sequence 𝐹 of extended real numbers, which is the infimum (indicated by <) of the set of suprema of all upper infinite subsequences of 𝐹. Definition 12-4.1 of [Gleason] p. 175. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 5-Sep-2014.) Obsolete version of limsupval 13915 as of 11-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝐹𝑉 → (lim sup‘𝐹) = sup(ran 𝐺, ℝ*, < ))

Theoremlimsupgf 13917* Closure of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       𝐺:ℝ⟶ℝ*

Theoremlimsupgval 13918* Value of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (𝑀 ∈ ℝ → (𝐺𝑀) = sup(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < ))

Theoremlimsupgle 13919* The defining property of the superior limit function. (Contributed by Mario Carneiro, 5-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ*) → ((𝐺𝐶) ≤ 𝐴 ↔ ∀𝑗𝐵 (𝐶𝑗 → (𝐹𝑗) ≤ 𝐴)))

Theoremlimsuple 13920* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺𝑗)))

TheoremlimsupleOLD 13921* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.) Obsolete version of limsuple 13920 as of 12-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺𝑗)))

Theoremlimsuplt 13922* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))       ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*𝐴 ∈ ℝ*) → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑗 ∈ ℝ (𝐺𝑗) < 𝐴))

Theoremlimsupval2 13923* The superior limit, relativized to an unbounded set. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    &   (𝜑𝐹𝑉)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)       (𝜑 → (lim sup‘𝐹) = inf((𝐺𝐴), ℝ*, < ))

Theoremlimsupgre 13924* If a sequence of real numbers has upper bounded limit supremum, then all the partial suprema are real. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    &   𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℝ ∧ (lim sup‘𝐹) < +∞) → 𝐺:ℝ⟶ℝ)

Theoremlimsupbnd1 13925* If a sequence is eventually at most 𝐴, then the limsup is also at most 𝐴. (The converse is only true if the less or equal is replaced by strictly less than; consider the sequence 1 / 𝑛 which is never less or equal to zero even though the limsup is.) (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐹:𝐵⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴))       (𝜑 → (lim sup‘𝐹) ≤ 𝐴)

Theoremlimsupbnd1OLD 13926* If a sequence is eventually at most 𝐴, then the limsup is also at most 𝐴. (The converse is only true if the less or equal is replaced by strictly less than; consider the sequence 1 / 𝑛 which is never less or equal to zero even though the limsup is.) (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.) Obsolete version of limsupbnd1 13925 as of 12-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐹:𝐵⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴))       (𝜑 → (lim sup‘𝐹) ≤ 𝐴)

Theoremlimsupbnd2 13927* If a sequence is eventually greater than 𝐴, then the limsup is also greater than 𝐴. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐹:𝐵⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑 → sup(𝐵, ℝ*, < ) = +∞)    &   (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)))       (𝜑𝐴 ≤ (lim sup‘𝐹))

Theoremlimsupbnd2OLD 13928* If a sequence is eventually greater than 𝐴, then the limsup is also greater than 𝐴. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.) Obsolete version of limsupbnd2 13927 as of 12-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑𝐹:𝐵⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑 → sup(𝐵, ℝ*, < ) = +∞)    &   (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)))       (𝜑𝐴 ≤ (lim sup‘𝐹))

5.10.2  Limits

Syntaxcli 13929 Extend class notation with convergence relation for limits.
class

Syntaxcrli 13930 Extend class notation with real convergence relation for limits.
class 𝑟

Syntaxco1 13931 Extend class notation with the set of all eventually bounded functions.
class 𝑂(1)

Syntaxclo1 13932 Extend class notation with the set of all eventually upper bounded functions.
class ≤𝑂(1)

Definitiondf-clim 13933* Define the limit relation for complex number sequences. See clim 13939 for its relational expression. (Contributed by NM, 28-Aug-2005.)
⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥))}

Definitiondf-rlim 13934* Define the limit relation for partial functions on the reals. See rlim 13940 for its relational expression. (Contributed by Mario Carneiro, 16-Sep-2014.)
𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}

Definitiondf-o1 13935* Define the set of eventually bounded functions. We don't bother to build the full conception of big-O notation, because we can represent any big-O in terms of 𝑂(1) and division, and any little-O in terms of a limit and division. We could also use limsup for this, but it only works on integer sequences, while this will work for real sequences or integer sequences. (Contributed by Mario Carneiro, 15-Sep-2014.)
𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}

Definitiondf-lo1 13936* Define the set of eventually upper bounded real functions. This fills a gap in 𝑂(1) coverage, to express statements like 𝑓(𝑥) ≤ 𝑔(𝑥) + 𝑂(𝑥) via (𝑥 ∈ ℝ+ ↦ (𝑓(𝑥) − 𝑔(𝑥)) / 𝑥) ∈ ≤𝑂(1). (Contributed by Mario Carneiro, 25-May-2016.)
≤𝑂(1) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ 𝑚}

Theoremclimrel 13937 The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
Rel ⇝

Theoremrlimrel 13938 The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.)
Rel ⇝𝑟

Theoremclim 13939* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. This means that for any real 𝑥, no matter how small, there always exists an integer 𝑗 such that the absolute difference of any later complex number in the sequence and the limit is less than 𝑥. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝜑𝐹𝑉)    &   ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))

Theoremrlim 13940* Express the predicate: The limit of complex number function 𝐹 is 𝐶, or 𝐹 converges to 𝐶, in the real sense. This means that for any real 𝑥, no matter how small, there always exists a number 𝑦 such that the absolute difference of any number in the function beyond 𝑦 and the limit is less than 𝑥. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑧𝐴) → (𝐹𝑧) = 𝐵)       (𝜑 → (𝐹𝑟 𝐶 ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥))))

Theoremrlim2 13941* Rewrite rlim 13940 for a mapping operation. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
(𝜑 → ∀𝑧𝐴 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))

Theoremrlim2lt 13942* Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝜑 → ∀𝑧𝐴 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦 < 𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))

Theoremrlim3 13943* Restrict the range of the domain bound to reals greater than some 𝐷 ∈ ℝ. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝜑 → ∀𝑧𝐴 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℝ)       (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ (𝐷[,)+∞)∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))

Theoremclimcl 13944 Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐹𝐴𝐴 ∈ ℂ)

Theoremrlimpm 13945 Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝐹𝑟 𝐴𝐹 ∈ (ℂ ↑pm ℝ))

Theoremrlimf 13946 Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)

Theoremrlimss 13947 Domain closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)

Theoremrlimcl 13948 Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐹𝑟 𝐴𝐴 ∈ ℂ)

Theoremclim2 13949* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 13939. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))

Theoremclim2c 13950* Express the predicate 𝐹 converges to 𝐴. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → (𝐹𝐴 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐵𝐴)) < 𝑥))

Theoremclim0 13951* Express the predicate 𝐹 converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘𝐵) < 𝑥)))

Theoremclim0c 13952* Express the predicate 𝐹 converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘𝐵) < 𝑥))

Theoremrlim0 13953* Express the predicate 𝐵(𝑧) converges to 0. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
(𝜑 → ∀𝑧𝐴 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 0 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘𝐵) < 𝑥)))

Theoremrlim0lt 13954* Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
(𝜑 → ∀𝑧𝐴 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 0 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦 < 𝑧 → (abs‘𝐵) < 𝑥)))

Theoremclimi 13955* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐶 ∈ ℝ+)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐹𝐴)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝐶))

Theoremclimi2 13956* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐶 ∈ ℝ+)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐹𝐴)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐵𝐴)) < 𝐶)

Theoremclimi0 13957* Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐶 ∈ ℝ+)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐹 ⇝ 0)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘𝐵) < 𝐶)

Theoremrlimi 13958* Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 28-Feb-2015.)
(𝜑 → ∀𝑧𝐴 𝐵𝑉)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → (𝑧𝐴𝐵) ⇝𝑟 𝐶)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑅))

Theoremrlimi2 13959* Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 12-May-2016.)
(𝜑 → ∀𝑧𝐴 𝐵𝑉)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → (𝑧𝐴𝐵) ⇝𝑟 𝐶)    &   (𝜑𝐷 ∈ ℝ)       (𝜑 → ∃𝑦 ∈ (𝐷[,)+∞)∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑅))

Theoremello1 13960* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
(𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))

Theoremello12 13961* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ 𝑚)))

Theoremello12r 13962* Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
(((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥𝐴 (𝐶𝑥 → (𝐹𝑥) ≤ 𝑀)) → 𝐹 ∈ ≤𝑂(1))

Theoremlo1f 13963 An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)
(𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ)

Theoremlo1dm 13964 An eventually upper bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 26-May-2016.)
(𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ)

Theoremlo1bdd 13965* The defining property of an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
((𝐹 ∈ ≤𝑂(1) ∧ 𝐹:𝐴⟶ℝ) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ 𝑚))

Theoremello1mpt 13966* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))

Theoremello1mpt2 13967* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ (𝐶[,)+∞)∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))

Theoremello1d 13968* Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝐴𝐶𝑥)) → 𝐵𝑀)       (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))

Theoremlo1bdd2 13969* If an eventually bounded function is bounded on every interval 𝐴 ∩ (-∞, 𝑦) by a function 𝑀(𝑦), then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))    &   ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶𝑦)) → 𝑀 ∈ ℝ)    &   (((𝜑𝑥𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶𝑦) ∧ 𝑥 < 𝑦)) → 𝐵𝑀)       (𝜑 → ∃𝑚 ∈ ℝ ∀𝑥𝐴 𝐵𝑚)

Theoremlo1bddrp 13970* Refine o1bdd2 13986 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))    &   ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶𝑦)) → 𝑀 ∈ ℝ)    &   (((𝜑𝑥𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶𝑦) ∧ 𝑥 < 𝑦)) → 𝐵𝑀)       (𝜑 → ∃𝑚 ∈ ℝ+𝑥𝐴 𝐵𝑚)

Theoremelo1 13971* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹𝑦)) ≤ 𝑚))

Theoremelo12 13972* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦𝐴 (𝑥𝑦 → (abs‘(𝐹𝑦)) ≤ 𝑚)))

Theoremelo12r 13973* Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
(((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥𝐴 (𝐶𝑥 → (abs‘(𝐹𝑥)) ≤ 𝑀)) → 𝐹 ∈ 𝑂(1))

Theoremo1f 13974 An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐹 ∈ 𝑂(1) → 𝐹:dom 𝐹⟶ℂ)

Theoremo1dm 13975 An eventually bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ)

Theoremo1bdd 13976* The defining property of an eventually bounded function. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐹 ∈ 𝑂(1) ∧ 𝐹:𝐴⟶ℂ) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦𝐴 (𝑥𝑦 → (abs‘(𝐹𝑦)) ≤ 𝑚))

Theoremlo1o1 13977 A function is eventually bounded iff its absolute value is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
(𝐹:𝐴⟶ℂ → (𝐹 ∈ 𝑂(1) ↔ (abs ∘ 𝐹) ∈ ≤𝑂(1)))

Theoremlo1o12 13978* A function is eventually bounded iff its absolute value is eventually upper bounded. (This function is useful for converting theorems about ≤𝑂(1) to 𝑂(1).) (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝑂(1) ↔ (𝑥𝐴 ↦ (abs‘𝐵)) ∈ ≤𝑂(1)))

Theoremelo1mpt 13979* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥 → (abs‘𝐵) ≤ 𝑚)))

Theoremelo1mpt2 13980* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 12-May-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝑂(1) ↔ ∃𝑦 ∈ (𝐶[,)+∞)∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥 → (abs‘𝐵) ≤ 𝑚)))

Theoremelo1d 13981* Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝐴𝐶𝑥)) → (abs‘𝐵) ≤ 𝑀)       (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))

Theoremo1lo1 13982* A real function is eventually bounded iff it is eventually lower bounded and eventually upper bounded. (Contributed by Mario Carneiro, 25-May-2016.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝑂(1) ↔ ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ∧ (𝑥𝐴 ↦ -𝐵) ∈ ≤𝑂(1))))

Theoremo1lo12 13983* A lower bounded real function is eventually bounded iff it is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝑀𝐵)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝑂(1) ↔ (𝑥𝐴𝐵) ∈ ≤𝑂(1)))

Theoremo1lo1d 13984* A real eventually bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))       (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))

Theoremicco1 13985* Derive eventual boundedness from separate upper and lower eventual bounds. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝜑𝐴 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑𝑁 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝐴𝐶𝑥)) → 𝐵 ∈ (𝑀[,]𝑁))       (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))

Theoremo1bdd2 13986* If an eventually bounded function is bounded on every interval 𝐴 ∩ (-∞, 𝑦) by a function 𝑀(𝑦), then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))    &   ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶𝑦)) → 𝑀 ∈ ℝ)    &   (((𝜑𝑥𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶𝑦) ∧ 𝑥 < 𝑦)) → (abs‘𝐵) ≤ 𝑀)       (𝜑 → ∃𝑚 ∈ ℝ ∀𝑥𝐴 (abs‘𝐵) ≤ 𝑚)

Theoremo1bddrp 13987* Refine o1bdd2 13986 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝑂(1))    &   ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶𝑦)) → 𝑀 ∈ ℝ)    &   (((𝜑𝑥𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶𝑦) ∧ 𝑥 < 𝑦)) → (abs‘𝐵) ≤ 𝑀)       (𝜑 → ∃𝑚 ∈ ℝ+𝑥𝐴 (abs‘𝐵) ≤ 𝑚)

Theoremclimconst 13988* An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)       (𝜑𝐹𝐴)

Theoremrlimconst 13989* A constant sequence converges to its value. (Contributed by Mario Carneiro, 16-Sep-2014.)
((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℂ) → (𝑥𝐴𝐵) ⇝𝑟 𝐵)

Theoremrlimclim1 13990 Forward direction of rlimclim 13991. (Contributed by Mario Carneiro, 16-Sep-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑟 𝐴)    &   (𝜑𝑍 ⊆ dom 𝐹)       (𝜑𝐹𝐴)

Theoremrlimclim 13991 A sequence on an upper integer set converges in the real sense iff it converges in the integer sense. (Contributed by Mario Carneiro, 16-Sep-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶ℂ)       (𝜑 → (𝐹𝑟 𝐴𝐹𝐴))

Theoremclimrlim2 13992* Produce a real limit from an integer limit, where the real function is only dependent on the integer part of 𝑥. (Contributed by Mario Carneiro, 2-May-2016.)
𝑍 = (ℤ𝑀)    &   (𝑛 = (⌊‘𝑥) → 𝐵 = 𝐶)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → (𝑛𝑍𝐵) ⇝ 𝐷)    &   ((𝜑𝑛𝑍) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝑀𝑥)       (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐷)

Theoremclimconst2 13993 A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
(ℤ𝑀) ⊆ 𝑍    &   𝑍 ∈ V       ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝐴}) ⇝ 𝐴)

Theoremclimz 13994 The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
(ℤ × {0}) ⇝ 0

Theoremrlimuni 13995 A real function whose domain is unbounded above converges to at most one limit. (Contributed by Mario Carneiro, 8-May-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   (𝜑𝐹𝑟 𝐵)    &   (𝜑𝐹𝑟 𝐶)       (𝜑𝐵 = 𝐶)

Theoremrlimdm 13996 Two ways to express that a function has a limit. (The expression ( ⇝𝑟𝐹) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Mario Carneiro, 8-May-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)       (𝜑 → (𝐹 ∈ dom ⇝𝑟𝐹𝑟 ( ⇝𝑟𝐹)))

Theoremclimuni 13997 An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
((𝐹𝐴𝐹𝐵) → 𝐴 = 𝐵)

Theoremfclim 13998 The limit relation is function-like, and with range the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
⇝ :dom ⇝ ⟶ℂ

Theoremclimdm 13999 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 Mario Carneiro, 18-Mar-2014.)
(𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))

Theoremclimeu 14000* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
(𝐹𝐴 → ∃!𝑥 𝐹𝑥)

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