Theorem xlimmnflimsup

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

Proof

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