In Stochastic analysis lecture notes P6 by Paul Bourgade, about a total variation of Brownian motion is infinite, he writes that
$$E\Big( \sum_{i=1}^m \vert B(t_{i+1})-B(t_{i})\vert \Big)\rightarrow\sum_{i=1}^m \sqrt{t_{i+1}-t_{i}}E(\mathscr{N}_{i})\geq E(\mathscr{N}_{1})\sqrt{t-s}\sqrt{m} \, (*)$$ where $B(t_{i+1})-B(t_{i})\to \sqrt{t_{i+1}-t_{i}}\mathscr{N}_{i}, $ $\mathscr{N}_{i}$ are independent standard Gaussian random variables.
But I have no idea how to use Cauchy-Schwarz inequality to show that the last inequality of $(*)$ is true.