Theorem xlimpnfliminf

Description: If a sequence of extended reals converges to +oo then its superior limit is also +oo . (Contributed by Glauco Siliprandi, 23-Apr-2023)

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

Proof

Step	Hyp	Ref	Expression
1		xlimpnfliminf.m	$$⊢\phi \to M\in \mathbb{Z}$$
2		xlimpnfliminf.z	$$⊢Z={\mathbb{Z}}_{\ge M}$$
3		xlimpnfliminf.f	$$⊢\phi \to F:Z⟶{ℝ}^{*}$$
4		xlimpnfliminf.c	$$⊢\phi \to F\underset{*}{⇝}\mathrm{+\infty }$$
5	1 2 3	xlimpnfv	$$⊢\phi \to \left(F\underset{*}{⇝}\mathrm{+\infty }\leftrightarrow \forall x\in ℝ\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}x\le F\left(j\right)\right)$$
6	4 5	mpbid	$$⊢\phi \to \forall x\in ℝ\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}x\le F\left(j\right)$$
7		nfcv	$$⊢\underset{\_}{Ⅎ}jF$$
8	7 1 2 3	liminfpnfuz	$$⊢\phi \to \left(\mathrm{lim\; inf}\left(F\right)=\mathrm{+\infty }\leftrightarrow \forall x\in ℝ\exists k\in Z\forall j\in {\mathbb{Z}}_{\ge k}x\le F\left(j\right)\right)$$
9	6 8	mpbird	$$⊢\phi \to \mathrm{lim\; inf}\left(F\right)=\mathrm{+\infty }$$