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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | climresmpt 42301* | 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 42302* | 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 42303* | 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 42304* | 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 42305* | 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 42306* | 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 42307* | 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 42308* | Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. Similar to clim 14843, 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 42309* | 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 42310* | 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 42311* | 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 42312* | Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 14843. Similar to clim2 14853, 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 42313* | 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 42314* | 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 42315* | 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 42316* | 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 42317* | 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 42318* | 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 42319* | 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 42320* | 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 42321* | 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 42322* | 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 42323* | 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 42324* | Two functions that are eventually equal to one another have the same limit. TODO: this is more general than climfveqmpt 42313 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 42325* | 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 42326* | 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 42327* | 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 42328* | 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 42329* | 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 42330* | 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 42331* | Two functions that are eventually equal, either both are convergent or both are divergent. TODO: this is more general than climeldmeqmpt 42310 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 42332 | Closure of the superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ*) | ||
Theorem | climfv 42333 | 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 42334* | 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 42335* | 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 42336 | The superior limit of the empty set. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (lim sup‘∅) = -∞ | ||
Theorem | climeldmeqmpt2 42337* | 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 42338 | 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 42339* | 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 42340 | 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 42341 | 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 42342 | 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 42343* | 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 42344* | 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 42345 | 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 42346* | 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 42347 | 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 42348* | 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 42349* | 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 42350* | 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 42351 | 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 42352* | 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 42353* | 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 42354* | 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 42355* | 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 42356* | 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 42357* | 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 42358* | 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 42359* | 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 42360* | 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 42361* | 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 42362* | 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 42363* | 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 42364* | 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 42365* | 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 42366* | 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 42367* | 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 42368* | 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 42369* | 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 42370* | 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 42371* | 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 42372* | 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 42373* | 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 42374* | 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 42375* | 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 42376* | 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 42377* | 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 42378* | 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 42379* | 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 42380* | 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 42381* | 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 42382* | 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 42383* | 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 42384 | If ∅ is a complex number, then it converges to itself. See 0ncn 10544 and its comment; see also the comment in climlimsupcex 42411. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) | ||
Theorem | climuzlem 42385* | 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 42386* | 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 42387* | 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 42388* | 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 42389* | 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 42390* | 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 42391* | 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 42392* | 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 42393 | 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 42394* | 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 42395* | 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 42396 | 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 42397* | 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 42398* | 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 42399 | Closure of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐹 ∈ 𝑉 ⇒ ⊢ (lim sup‘𝐹) ∈ ℝ* | ||
Theorem | liminfgf 42400 | Closure of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⇒ ⊢ 𝐺:ℝ⟶ℝ* |
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