first hitting time of drift brownian motion is stopping time?

Question

$B_t$ be an ($\mathscr{F}_t$)-Brownian motion started from $0$. $a>0$ and define $$\sigma_a = \inf \{t \geq 0 : B_t \leq t-a \}. $$

I want to show that $\sigma_a$ is a stopping time and that $\sigma_a < \infty$ a.s.

For prove second part, I think that I can use $\frac{B_t -t}{t} \to -1 $ a.s. is it right?

Also I need some help for first part... Thanks.