P. 462 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 46101-46200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	limclr 46101	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)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))))    &   ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞))))    &   ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵))    &   ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵))    ⇒   ⊢ (𝜑 → (((𝐹 limℂ 𝐵) ≠ ∅ ↔ 𝐿 = 𝑅) ∧ (𝐿 = 𝑅 → 𝐿 ∈ (𝐹 limℂ 𝐵))))
Theorem	divlimc 46102*	Limit of the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 / 𝐶))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (ℂ ∖ {0}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐹 limℂ 𝐷))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐺 limℂ 𝐷))    &   ⊢ (𝜑 → 𝑌 ≠ 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → (𝑋 / 𝑌) ∈ (𝐻 limℂ 𝐷))
Theorem	expfac 46103*	Factorial grows faster than exponential. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ⇝ 0)
Theorem	climconstmpt 46104*	A constant sequence converges to its value. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴) ⇝ 𝐴)
Theorem	climresmpt 46105*	A function restricted to upper integers converges iff the original function converges. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑍 ↦ 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ 𝐺 = (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴)    ⇒   ⊢ (𝜑 → (𝐺 ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵))
Theorem	climsubmpt 46106*	Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴) ⇝ 𝐶)    &   ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐷)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐴 − 𝐵)) ⇝ (𝐶 − 𝐷))
Theorem	climsubc2mpt 46107*	Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴) ⇝ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐴 − 𝐵)) ⇝ (𝐶 − 𝐵))
Theorem	climsubc1mpt 46108*	Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝐴 − 𝐵)) ⇝ (𝐴 − 𝐶))
Theorem	fnlimfv 46109*	The value of the limit function 𝐺 at any point of its domain 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐷    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝐺‘𝑋) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
Theorem	climreclf 46110*	The limit of a convergent real sequence is real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	climeldmeq 46111*	Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ))
Theorem	climf2 46112*	Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. Similar to clim 15447, but without the disjoint var constraint 𝜑𝑘 and 𝐹𝑘. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))))
Theorem	fnlimcnv 46113*	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.)
⊢ Ⅎ𝑥𝐹    &   ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }    &   ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ (𝐺‘𝑋))
Theorem	climeldmeqmpt 46114*	Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐷 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ dom ⇝ ↔ (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ dom ⇝ ))
Theorem	climfveq 46115*	Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺))
Theorem	clim2f2 46116*	Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 15447. Similar to clim2 15457, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))))
Theorem	climfveqmpt 46117*	Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐷 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → ( ⇝ ‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = ( ⇝ ‘(𝑘 ∈ 𝐶 ↦ 𝐷)))
Theorem	climd 46118*	Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋))
Theorem	clim2d 46119*	The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋))
Theorem	fnlimfvre 46120*	The limit function of real functions, applied to elements in its domain, evaluates to Real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑚𝜑    &   ⊢ Ⅎ𝑚𝐹    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)    &   ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
Theorem	allbutfifvre 46121*	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.)
⊢ Ⅎ𝑚𝜑    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)    &   ⊢ 𝐷 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ)
Theorem	climleltrp 46122*	The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
Theorem	fnlimfvre2 46123*	The limit function of real functions, applied to elements in its domain, evaluates to Real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑚𝜑    &   ⊢ Ⅎ𝑚𝐹    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)    &   ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }    &   ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝐺‘𝑋) ∈ ℝ)
Theorem	fnlimf 46124*	The limit function of real functions, is a real-valued function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑚𝜑    &   ⊢ Ⅎ𝑚𝐹    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)    &   ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }    &   ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))    ⇒   ⊢ (𝜑 → 𝐺:𝐷⟶ℝ)
Theorem	fnlimabslt 46125*	A sequence of function values, approximates the corresponding limit function value, all but finitely many times. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑚𝜑    &   ⊢ Ⅎ𝑚𝐹    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)    &   ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }    &   ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)(((𝐹‘𝑚)‘𝑋) ∈ ℝ ∧ (abs‘(((𝐹‘𝑚)‘𝑋) − (𝐺‘𝑋))) < 𝑌))
Theorem	climfveqf 46126*	Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ Ⅎ𝑘𝐺    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺))
Theorem	climmptf 46127*	Exhibit a function 𝐺 with the same convergence properties as the not-quite-function 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐺 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴))
Theorem	climfveqmpt3 46128*	Two functions that are eventually equal to one another have the same limit. TODO: this is more general than climfveqmpt 46117 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → ( ⇝ ‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = ( ⇝ ‘(𝑘 ∈ 𝐶 ↦ 𝐷)))
Theorem	climeldmeqf 46129*	Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ Ⅎ𝑘𝐺    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ))
Theorem	climreclmpt 46130*	The limit of B convergent real sequence is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴) ⇝ 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ∈ ℝ)
Theorem	limsupref 46131*	If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
Theorem	limsupbnd1f 46132*	If a sequence is eventually at most 𝐴, then the limsup is also at most 𝐴. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴)
Theorem	climbddf 46133*	A converging sequence of complex numbers is bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥)
Theorem	climeqf 46134*	Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ Ⅎ𝑘𝐺    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴))
Theorem	climeldmeqmpt3 46135*	Two functions that are eventually equal, either both are convergent or both are divergent. TODO: this is more general than climeldmeqmpt 46114 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ dom ⇝ ↔ (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ dom ⇝ ))
Theorem	limsupcld 46136	Closure of the superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
Theorem	climfv 46137	The limit of a convergent sequence, expressed as the function value of the convergence relation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐹 ⇝ 𝐴 → 𝐴 = ( ⇝ ‘𝐹))
Theorem	limsupval3 46138*	The superior limit of an infinite sequence 𝐹 of extended real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    &   ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
Theorem	climfveqmpt2 46139*	Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ( ⇝ ‘(𝑘 ∈ 𝐴 ↦ 𝐶)) = ( ⇝ ‘(𝑘 ∈ 𝐵 ↦ 𝐶)))
Theorem	limsup0 46140	The superior limit of the empty set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (lim sup‘∅) = -∞
Theorem	climeldmeqmpt2 46141*	Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∈ dom ⇝ ↔ (𝑘 ∈ 𝐵 ↦ 𝐶) ∈ dom ⇝ ))
Theorem	limsupresre 46142	The supremum limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (lim sup‘(𝐹 ↾ ℝ)) = (lim sup‘𝐹))
Theorem	climeqmpt 46143*	Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝ 𝐷 ↔ (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝ 𝐷))
Theorem	climfvd 46144	The limit of a convergent sequence, expressed as the function value of the convergence relation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹 ⇝ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = ( ⇝ ‘𝐹))
Theorem	limsuplesup 46145	An upper bound for the superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐾 ∈ ℝ)    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) ≤ sup(((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*), ℝ*, < ))
Theorem	limsupresico 46146	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.)
⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ 𝑍 = (𝑀[,)+∞)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝑍)) = (lim sup‘𝐹))
Theorem	limsuppnfdlem 46147*	If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))    &   ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) = +∞)
Theorem	limsuppnfd 46148*	If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) = +∞)
Theorem	limsupresuz 46149	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.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ)    ⇒   ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝑍)) = (lim sup‘𝐹))
Theorem	limsupub 46150*	If the limsup is not +∞, then the function is eventually bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝜑    &   ⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    &   ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
Theorem	limsupres 46151	The superior limit of a restriction is less than or equal to the original superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝐶)) ≤ (lim sup‘𝐹))
Theorem	climinf2lem 46152*	A convergent, nonincreasing sequence, converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < ))
Theorem	climinf2 46153*	A convergent, nonincreasing sequence, converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < ))
Theorem	limsupvaluz 46154*	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.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ*, < ))
Theorem	limsupresuz2 46155	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.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → dom 𝐹 ⊆ ℤ)    ⇒   ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝑍)) = (lim sup‘𝐹))
Theorem	limsuppnflem 46156*	If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
Theorem	limsuppnf 46157*	If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
Theorem	limsupubuzlem 46158*	If the limsup is not +∞, then the function is bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝜑    &   ⊢ Ⅎ𝑗𝑋    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝐾 ∈ ℝ)    &   ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 (𝐾 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑌))    &   ⊢ 𝑁 = if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾))    &   ⊢ 𝑊 = sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < )    &   ⊢ 𝑋 = if(𝑊 ≤ 𝑌, 𝑌, 𝑊)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
Theorem	limsupubuz 46159*	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.)
⊢ Ⅎ𝑗𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
Theorem	climinf2mpt 46160*	A bounded below, monotonic nonincreasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑗𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)    &   ⊢ (𝑘 = 𝑗 → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ 𝑗 = (𝑘 + 1)) → 𝐶 ≤ 𝐵)    &   ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ⇝ inf(ran (𝑘 ∈ 𝑍 ↦ 𝐵), ℝ*, < ))
Theorem	climinfmpt 46161*	A bounded below, monotonic nonincreasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑗𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)    &   ⊢ (𝑘 = 𝑗 → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ 𝑗 = (𝑘 + 1)) → 𝐶 ≤ 𝐵)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ⇝ inf(ran (𝑘 ∈ 𝑍 ↦ 𝐵), ℝ*, < ))
Theorem	climinf3 46162*	A convergent, nonincreasing sequence, converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))    &   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < ))
Theorem	limsupvaluzmpt 46163*	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.)
⊢ Ⅎ𝑗𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝑍 ↦ 𝐵)) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝑘) ↦ 𝐵), ℝ*, < )), ℝ*, < ))
Theorem	limsupequzmpt2 46164*	Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝜑    &   ⊢ Ⅎ𝑗𝐴    &   ⊢ Ⅎ𝑗𝐵    &   ⊢ 𝐴 = (ℤ≥‘𝑀)    &   ⊢ 𝐵 = (ℤ≥‘𝑁)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)))
Theorem	limsupubuzmpt 46165*	If the limsup is not +∞, then the function is eventually bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝜑    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝑍 ↦ 𝐵)) ≠ +∞)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥)
Theorem	limsupmnflem 46166*	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.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    &   ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
Theorem	limsupmnf 46167*	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.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
Theorem	limsupequzlem 46168*	Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 Fn (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺 Fn (ℤ≥‘𝑁))    &   ⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺))
Theorem	limsupequz 46169*	Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ Ⅎ𝑘𝐺    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 Fn (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺 Fn (ℤ≥‘𝑁))    &   ⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺))
Theorem	limsupre2lem 46170*	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.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥))))
Theorem	limsupre2 46171*	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.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥))))
Theorem	limsupmnfuzlem 46172*	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.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
Theorem	limsupmnfuz 46173*	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.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
Theorem	limsupequzmptlem 46174*	Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ 𝐴 = (ℤ≥‘𝑀)    &   ⊢ 𝐵 = (ℤ≥‘𝑁)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ 𝑊)    &   ⊢ 𝐾 = if(𝑀 ≤ 𝑁, 𝑁, 𝑀)    ⇒   ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)))
Theorem	limsupequzmpt 46175*	Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ 𝐴 = (ℤ≥‘𝑀)    &   ⊢ 𝐵 = (ℤ≥‘𝑁)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)))
Theorem	limsupre2mpt 46176*	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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 < 𝐵) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 < 𝑦))))
Theorem	limsupequzmptf 46177*	Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝜑    &   ⊢ Ⅎ𝑗𝐴    &   ⊢ Ⅎ𝑗𝐵    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ 𝐴 = (ℤ≥‘𝑀)    &   ⊢ 𝐵 = (ℤ≥‘𝑁)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)))
Theorem	limsupre3lem 46178*	Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))))
Theorem	limsupre3 46179*	Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))))
Theorem	limsupre3mpt 46180*	Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦))))
Theorem	limsupre3uzlem 46181*	Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, infinitely often; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)))
Theorem	limsupre3uz 46182*	Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, infinitely often; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)))
Theorem	limsupreuz 46183*	Given a function on the reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, infinitely often; 2. there is a real number that is greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝐹    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    ⇒   ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)))
Theorem	limsupvaluz2 46184*	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.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ, < ))
Theorem	limsupreuzmpt 46185*	Given a function on the reals, defined on a set of upper integers, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, infinitely often; 2. there is a real number that is greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑗𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((lim sup‘(𝑗 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ 𝐵 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥)))
Theorem	supcnvlimsup 46186*	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 46187*	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 46188	If ∅ is a complex number, then it converges to itself. See 0ncn 11047 and its comment; see also the comment in climlimsupcex 46215. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (∅ ∈ ℂ → ∅ ⇝ ∅)
Theorem	climuzlem 46189*	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 46190*	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 46191*	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 46192*	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 46193*	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 46194*	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 46195*	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. (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 46196*	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)
21.43.7.1  Inferior limit (lim inf)
Syntax	clsi 46197	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 46198*	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 46199*	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 46200	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*, < ))