 Solution

The scaled process starts at zero and has continuous paths. Disjoint time intervals map to disjoint intervals under $t\mapsto c^{2}t$, preserving independence of increments, andThis holds for either sign of $c\ne 0$. Hence $\boxed{{\tilde{B}}_t=c^{-1}{B}_{c^{2}t}\text{ is standard Brownian motion}}$.