Solution
= Solution
A radial timelike geodesic has conserved $E=f\dot t$ and normalization $\dot r^2=E^2-f$. Choose $E>1$ and the outward sign. Then
$$
\Delta\tau=\int_0^b\frac{dr}{\sqrt{E^2-1+r^2/b^2}}<\infty,
$$
so the observer reaches $r=b$ in finite <proper time> even though static coordinate $t$ diverges.