Theorem liminfcld

Description: Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022)

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

Proof

Step	Hyp	Ref	Expression
1		liminfcld.1	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
2		liminfcl	⊢ ( 𝐹 ∈ 𝑉 → ( lim inf ‘ 𝐹 ) ∈ ℝ* )
3	1 2	syl	⊢ ( 𝜑 → ( lim inf ‘ 𝐹 ) ∈ ℝ* )