I am working on the following problem:
Show that if $B_t - B_s, 0 \leq s < t,$ is normally distributed with mean zero and variance $t-s$, then for each positive integer $n$ there is a positive constant $C_n$ such that $$ E|B_t - B_s|^{2n}=C_n|t-s|^n $$
I was thinking about trying to use induction. If $n=1$ we have $E|B_t-B_s|^2 = t-s=|t-s|.$
So if I suppose it holds for $n-1$ I want to show it holds for $n$. However, I can't find any way that it holding for $n-1$ makes it hold for $n$.
We're currently learning about Brownian motion so I'm thinking I somehow need use the properties of Brownian motion, but I can't find any helpful properties in Durrett.
Any help would be appreciated.