Is the following stopping time finite: $T:=\inf\{t\geq 0:B_t\geq \sqrt{t}+1\}?$

Question

We have a Brownian motion process $B$ and a stopping time defined like this: $$T:=\inf\{t\geq 0:B_t\geq \sqrt{t}+1\}.$$

Is this stopping time almost surely finite, eg. $T<\infty$, and why?

My intuition would say it is.