Theorem climliminflimsup4

Description: A sequence of real numbers converges if and only if its superior limit is real and equal to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypotheses	climliminflimsup4.1	$$⊢\phi \to M\in \mathbb{Z}$$
	climliminflimsup4.2	$$⊢Z={\mathbb{Z}}_{\ge M}$$
	climliminflimsup4.3	$$⊢\phi \to F:Z⟶ℝ$$
Assertion	climliminflimsup4	$$⊢\phi \to \left(F\in \mathrm{dom}⇝\leftrightarrow \left(\mathrm{lim\; sup}\left(F\right)\in ℝ\wedge \mathrm{lim\; inf}\left(F\right)=\mathrm{lim\; sup}\left(F\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		climliminflimsup4.1	$$⊢\phi \to M\in \mathbb{Z}$$
2		climliminflimsup4.2	$$⊢Z={\mathbb{Z}}_{\ge M}$$
3		climliminflimsup4.3	$$⊢\phi \to F:Z⟶ℝ$$
4	1 2 3	climliminflimsup2	$$⊢\phi \to \left(F\in \mathrm{dom}⇝\leftrightarrow \left(\mathrm{lim\; sup}\left(F\right)\in ℝ\wedge \mathrm{lim\; sup}\left(F\right)\le \mathrm{lim\; inf}\left(F\right)\right)\right)$$
5	3	frexr	$$⊢\phi \to F:Z⟶{ℝ}^{*}$$
6	1 2 5	liminfgelimsupuz	$$⊢\phi \to \left(\mathrm{lim\; sup}\left(F\right)\le \mathrm{lim\; inf}\left(F\right)\leftrightarrow \mathrm{lim\; inf}\left(F\right)=\mathrm{lim\; sup}\left(F\right)\right)$$
7	6	anbi2d	$$⊢\phi \to \left(\left(\mathrm{lim\; sup}\left(F\right)\in ℝ\wedge \mathrm{lim\; sup}\left(F\right)\le \mathrm{lim\; inf}\left(F\right)\right)\leftrightarrow \left(\mathrm{lim\; sup}\left(F\right)\in ℝ\wedge \mathrm{lim\; inf}\left(F\right)=\mathrm{lim\; sup}\left(F\right)\right)\right)$$
8	4 7	bitrd	$$⊢\phi \to \left(F\in \mathrm{dom}⇝\leftrightarrow \left(\mathrm{lim\; sup}\left(F\right)\in ℝ\wedge \mathrm{lim\; inf}\left(F\right)=\mathrm{lim\; sup}\left(F\right)\right)\right)$$