Theorem limsupcld

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

	Ref	Expression
Hypothesis	limsupcld.1	$$⊢\phi \to F\in V$$
Assertion	limsupcld	$$⊢\phi \to \mathrm{lim\; sup}\left(F\right)\in {ℝ}^{*}$$

Proof

Step	Hyp	Ref	Expression
1		limsupcld.1	$$⊢\phi \to F\in V$$
2		limsupcl	$$⊢F\in V\to \mathrm{lim\; sup}\left(F\right)\in {ℝ}^{*}$$
3	1 2	syl	$$⊢\phi \to \mathrm{lim\; sup}\left(F\right)\in {ℝ}^{*}$$