Is this an ok proof for a.s no interval of monotonicity for sample paths of B.M
EDIT : [In the proceeding proof I just prove the sample path is not strictly increasing, it is an equivalent proof to show it is not strictly decreasing ]
first we show for the interval [0,1]
$P(B(t)-B(s)\geq 0$ $ \forall s,t\in [0,1]) $
$\leq P(B(\frac{i+1}{n})-B(\frac{i}{n})\geq0 $ for $i =0,...n-1$ )= $\frac{1}{2^n}$ $\to 0 $
now we use scaling properties and stationary increments to prove for rational intervals [p,q]
consider for , p,q rational
$P(B(t)-B(s)\geq 0$ $ \forall s,t\in [p,q]) $
$=P(B(t-p)-B(s-p)\geq 0$ $ \forall s,t\in [p,q]) $ by stationary property
$=P(B(t)-B(s)\geq 0$ $ \forall s,t\in [0,q-p]) $
$=P((\frac{1}{q-p})^{-1/2}B(t\frac{1}{q-p})-(\frac{1}{q-p})^{-1/2}B(s\frac{1}{q-p})\geq 0$ $ \forall s,t\in [0,q-p])$ by scaling property
$=P(B(t)-B(s)\geq 0$ $ \forall s,t\in [0,1]) $
edit: If we have no monitancity in all rational intervals then we have no monotonicity in any arbitrary interval since all rational intervals contain such arbitrary intervals.