The Brownian motion $\{W_t: t \geq 0\}$ satisfies the independent increment property:
If $0 \leq t_0 < t_1 <\dots < t_n$, then $$W_{t_0}, W_{t_1}-W_{t_0}, \dots, W_{t_n}-W_{t_{n-1}}$$
are independent.
Why is there a strict inequality in the $0 \leq t_0 < t_1 < \dots < t_n$?
If $t_i \leq t_{i+1}$ for some $i$, then we just have
$$W_{t_{i+1}}-W_{t_i} = 0$$ which will not affect the independency, as a constant random variable is independent from all other random variables.
What am I missing?