Below is the proof of the translation invariance of a one dimensional Brownian motion starting at the origin, given by Durrett in Probability Theory and Examples. However, the proof only shows independence, and I don't see how $\{B_t-B_0:t\ge 0\}$ has the same distribution as a Brownian motion with $B_0=0$ as stated in the theorem. I would greatly appreciate it if anyone could explain this part to me.
