Theorem limsupcld

Description: Closure of the superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypothesis	limsupcld.1	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
Assertion	limsupcld	⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) ∈ ℝ* )

Proof

Step	Hyp	Ref	Expression
1		limsupcld.1	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
2		limsupcl	⊢ ( 𝐹 ∈ 𝑉 → ( lim sup ‘ 𝐹 ) ∈ ℝ* )
3	1 2	syl	⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) ∈ ℝ* )