Theorem xlimpnfliminf2

Description: A sequence of extended reals converges to +oo if and only if its superior limit is also +oo . (Contributed by Glauco Siliprandi, 23-Apr-2023)

	Ref	Expression
Hypotheses	xlimpnfliminf2.m	$$⊢\phi \to M\in \mathbb{Z}$$
	xlimpnfliminf2.z	$$⊢Z={\mathbb{Z}}_{\ge M}$$
	xlimpnfliminf2.f	$$⊢\phi \to F:Z⟶{ℝ}^{*}$$
Assertion	xlimpnfliminf2	$$⊢\phi \to \left(F\underset{*}{⇝}\mathrm{+\infty }\leftrightarrow \mathrm{lim\; inf}\left(F\right)=\mathrm{+\infty }\right)$$

Proof

Step	Hyp	Ref	Expression
1		xlimpnfliminf2.m	$$⊢\phi \to M\in \mathbb{Z}$$
2		xlimpnfliminf2.z	$$⊢Z={\mathbb{Z}}_{\ge M}$$
3		xlimpnfliminf2.f	$$⊢\phi \to F:Z⟶{ℝ}^{*}$$
4	1 2 3	xlimpnfv	$$⊢\phi \to \left(F\underset{*}{⇝}\mathrm{+\infty }\leftrightarrow \forall x\in ℝ\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}x\le F\left(j\right)\right)$$
5		nfcv	$$⊢\underset{\_}{Ⅎ}jF$$
6	5 1 2 3	liminfpnfuz	$$⊢\phi \to \left(\mathrm{lim\; inf}\left(F\right)=\mathrm{+\infty }\leftrightarrow \forall x\in ℝ\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}x\le F\left(j\right)\right)$$
7	4 6	bitr4d	$$⊢\phi \to \left(F\underset{*}{⇝}\mathrm{+\infty }\leftrightarrow \mathrm{lim\; inf}\left(F\right)=\mathrm{+\infty }\right)$$