There are many definitions of Brownian Motion and I am now working on the following:
A collection of random variables $B : [0, ∞)×Ω → R$ is called a Brownian motion if
(a) $B(·, ω)$ is continuous and $B(0, ω) = 0$ for any $ω ∈ Ω$
(b) Any non-overlapping increments $B(a) − B(b)$, $B(c) − B(d)$ are independent, where $a < b ≤c < d$.
(c) $B(t + s) − B(s) ∼ N (0, t)$ for any $t > 0, s ≥ 0$
Then I was asked to show that $$B(t_1) − B(0), B(t_2) − B(t_1), · · · , B(t_n) − B(t_{n−1})$$ are independent.
This is weird since I believe that their (mutual) indenpendence are actually followed from the definition. So how should I start?