Proper-time bound inside a Schwarzschild black hole ($\Delta s\le 2M\arcsin \sqrt{r_0/(2M)}-\sqrt{r_0(2M-r_0)}<\pi M$)

Inside the future black-hole region, the radius decreases along every future-directed timelike path. Its unit tangent obeys $|dr/ds|\ge \sqrt{2M/r-1}$ from the metric normalization, independently of radial acceleration or angular motion. Integrating from $r_0<2M$ to zero gives the displayed finite upper bound. The statement concerns future continuation within the black-hole region; reversing time in the white-hole region reverses the radial direction.