Let $X_t=2+t+W_t$ for $t\ge 0$, where $(W_t)_{t\ge 0}$ is a standard Brownian motion. For every $n\ge 1$, set $X^n_t:=X_t-{\bf 1}_{t\ge n}$. Denote respectively

$$\tau:=\inf\{t\ge 0:~ X_t\le 0\}\quad \mbox{and} \quad \tau^n:=\inf\{t\ge 0:~ X^n_t\le 0\}.$$

Could we prove or disprove $\lim_{n\to\infty}\mathbb P[\tau^n=\infty]=\mathbb P[\tau=\infty]$?

**Further Question** : I wish to prove the similar convergence result. This question can be found at Convergence of the probabilities that drifted Brownian motion with jump never hits zero (continuation)