 Solution

Fix rational $0<p<q$. Divide $[p,q]$ into $N$ equal intervals. Its Brownian increments are independent centered normal variables, so the probability that all $N$ are nonnegative is $2^{-N}$. A nondecreasing path would force this event for every $N$, hence its probability is zero. The same argument with nonpositive increments excludes a nonincreasing path.

There are countably many rational pairs $p<q$, so with probability one none of these intervals supports a monotone path. Every real interval $0<a<b<\infty$ contains such a rational subinterval. Monotonicity on the larger interval would imply monotonicity on that subinterval, a contradiction. Thus the nowhere monotonicity of Brownian motion assertion holds simultaneously:Both nondecreasing and nonincreasing behavior, including a constant path segment, are excluded.