I'm confused with an inequality arising in the proof of Proposition 2.1 in Cucchiero, Rigger, and Svaluto-Ferro - Propagation of minimality in the supercooled Stefan problem (page 10).

Let $\ell^n: [0,\infty)\to [0,1]$ be right-continuous and increasing functions s.t. $\ell^n(0)=0$. Given $x>0$ and Brownian motion $(B_t)_{t\ge 0}$, the authors claim that reverse Fatou's lemma implies the following inequality $$\limsup_{n\to\infty}\mathbb P[\exists s\in [0,t]:~ x+B_s\le \ell^n(s)]\le \mathbb P[\exists s\in [0,t]:~ x+B_s\le \limsup_{n\to\infty}\ell^n(s)],\quad \forall t>0.$$

But I do not see why the inequality $$\limsup_{n\to\infty} {\bf 1}_{\{\exists s\in [0,t]: x+B_s\le \ell^n(s)\}}\le {\bf 1}_{\{\exists s\in [0,t]: x+B_s\le \limsup_{n\to\infty}\ell^n(s)\}}$$ holds. Is this obvious to everyone? Any explanation is highly appreciated!

Let $\ell^n: [0,\infty)\to [0,1]$ be right-continuous and increasing functions s.t. $\ell^n(0)=0$. Given $x>0$ and Brownian motion $(B_t)_{t\ge 0}$, can we prove $$\limsup_{n\to\infty}\mathbb P[\exists s\in [0,t]:~ x+B_s\le \ell^n(s)]\le \mathbb P[\exists s\in [0,t]:~ x+B_s\le \limsup_{n\to\infty}\ell^n(s)],\quad \forall t>0?$$

PS : It appears that the pathwise inequality $$\limsup_{n\to\infty} {\bf 1}_{\{\exists s\in [0,t]: x+B_s\le \ell^n(s)\}}\le {\bf 1}_{\{\exists s\in [0,t]: x+B_s\le \limsup_{n\to\infty}\ell^n(s)\}}$$ does not hold.

I'm confused with an inequality arising in the proof of Proposition 2.1 in https://arxiv.org/pdf/2010.03580.pdf (page 10).

Let $\ell^n: [0,\infty)\to [0,1]$ be right-continuous and increasing functions s.t. $\ell^n(0)=0$. Given $x>0$ and Brownian motion $(B_t)_{t\ge 0}$, the authors claim that reverse Fatou's lemma implies the following inequality

 $$\limsup_{n\to\infty}\mathbb P[\exists s\in [0,t]:~ x+B_s\le \ell^n(s)]\le \mathbb P[\exists s\in [0,t]:~ x+B_s\le \limsup_{n\to\infty}\ell^n(s)],\quad \forall t>0.$$

But I do not see why the inequality

 $$\limsup_{n\to\infty} {\bf 1}_{\{\exists s\in [0,t]: x+B_s\le \ell^n(s)\}}\le {\bf 1}_{\{\exists s\in [0,t]: x+B_s\le \limsup_{n\to\infty}\ell^n(s)\}}$$

holds. Is this obvious to everyone? Any explanation is highly appreciated!

I'm confused with an inequality arising in the proof of Proposition 2.1 in Cucchiero, Rigger, and Svaluto-Ferro - Propagation of minimality in the supercooled Stefan problem (page 10).

Let $\ell^n: [0,\infty)\to [0,1]$ be right-continuous and increasing functions s.t. $\ell^n(0)=0$. Given $x>0$ and Brownian motion $(B_t)_{t\ge 0}$, the authors claim that reverse Fatou's lemma implies the following inequality  $$\limsup_{n\to\infty}\mathbb P[\exists s\in [0,t]:~ x+B_s\le \ell^n(s)]\le \mathbb P[\exists s\in [0,t]:~ x+B_s\le \limsup_{n\to\infty}\ell^n(s)],\quad \forall t>0.$$

But I do not see why the inequality  $$\limsup_{n\to\infty} {\bf 1}_{\{\exists s\in [0,t]: x+B_s\le \ell^n(s)\}}\le {\bf 1}_{\{\exists s\in [0,t]: x+B_s\le \limsup_{n\to\infty}\ell^n(s)\}}$$ holds. Is this obvious to everyone? Any explanation is highly appreciated!