I am stuck on this issue:
Let $(B_t)$ be a Brownian motion. We know that since $\{0\}$ is a closed set in $\mathbb{R}$ and that $(B_t)$ is a continuous adapted process, $$ \tau:= \inf \{ t\geq 0 : B_t =0 \} $$ is a stopping time. However, I am not sure whether $$ \tau':= \inf \{ t\geq 1 : B_t =0 \} $$ is a stopping time. Intuitively, it should be, as it is the first zero after $1$.
But is there any result/theorem to justify this fact rigorously?
Let $X_t=(B_t,t)$ and $A=\{0\}\times[1,+\infty)$ then $\tau'$ is the first hitting time of $A$ by $X$, $A$ is closed in $\mathbb R^2$ and the process $X$ is continuous adapted. According to your question, you know that such hitting times are stopping times hence you are done.