Let $(f_n)_{n\ge 1}$ be a sequence of non-decreasing and continuous functions defined on $\mathbb R_+$ and taking values in $[0,1]$. Further, for each $t\ge 0$, $n\mapsto f_n(t)$ is non-decreasing. Denote by $f$ the point-wise limit of $(f_n)_{n\ge 1}$ and by $W$ a standard Brownian motion. For every $x>0$ (saying $x=1$), can we prove that the following convergence
$$\lim_{n\to\infty}\mathbb P[\inf_{0\le s\le t}(x+W_s-f_n(s))\le 0] = \mathbb P[\inf_{0\le s\le t}(x+W_s-f(s))\le 0]$$
holds for every $t\ge 0$?
PS : The direction $$\lim_{n\to\infty}\mathbb P[\inf_{0\le s\le t}(W_s-f_n(s))\le -x] \le \mathbb P[\inf_{0\le s\le t}(W_s-f(s))\le -x]$$ is trivial. For the other direction, my idea is as follows : Denote by $\tau_n$ (resp. $\tau$) the first hitting time of $W-f_n$ (resp. $W-f$) at $-x$. Then
$$\mathbb P[\inf_{0\le s\le t}(W_s-f(s))\le -x]-\mathbb P[\inf_{0\le s\le t}(W_s-f_n(s))\le -x] = \mathbb P[\tau\le t, \tau_n>t]$$
as $\tau\le \tau_n$. Rewrite
\begin{eqnarray} \mathbb P[\tau\le t, \tau_n>t] &=& \int_{[0,t]}\mathbb P[\tau_n>t|\tau=s]\mathbb P[\tau\in ds]\\ &=&\int_{[0,t)}\mathbb P[\tau_n>t|\tau=s]\mathbb P[\tau\in ds] + \mathbb P[\tau_n>t|\tau=t]\mathbb P[\tau=t]. \end{eqnarray}
How to estimate $\mathbb P[\tau_n>t|\tau=s]$? Is $\mathbb P[\tau=t]=0$?