(a) What is a Brownian motion?

(b) Let $\left(B_{t}, t \geqslant 0\right)$ be a Brownian motion. Show that the process $\tilde{B}_{t}:=\frac{1}{c} B_{c^{2} t}$, $c \in \mathbb{R} \backslash\{0\}$, is also a Brownian motion.

(c) Let $Z:=\sup _{t \geqslant 0} B_{t}$. Show that $c Z \stackrel{(d)}{=} Z$ for all $c>0$ (i.e. $c Z$ and $Z$ have the same laws). Conclude that $Z \in\{0,+\infty\}$ a.s.

(d) Show that $\mathbb{P}[Z=+\infty]=1$.