Description: 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)
Ref | Expression | ||
---|---|---|---|
Hypotheses | limsupvald.1 | ⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) | |
limsupvald.2 | ⊢ 𝐺 = ( 𝑘 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑘 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) | ||
Assertion | limsupvald | ⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) = inf ( ran 𝐺 , ℝ* , < ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupvald.1 | ⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) | |
2 | limsupvald.2 | ⊢ 𝐺 = ( 𝑘 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑘 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) | |
3 | 2 | limsupval | ⊢ ( 𝐹 ∈ 𝑉 → ( lim sup ‘ 𝐹 ) = inf ( ran 𝐺 , ℝ* , < ) ) |
4 | 1 3 | syl | ⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) = inf ( ran 𝐺 , ℝ* , < ) ) |