[a, b]

with rational endpoints and

a < b

, the

2^{m}

increments over its equal subdivision are independent centered normal random variables. If the Brownian motion path were nondecreasing, all these increments would be nonnegative, an event with probability

2^{- 2^{m}}

. Letting

m \to \infty

gives probability zero. The same argument excludes nonincreasing paths. A countable union over rational intervals shows that, almost surely, no nontrivial interval supports a monotone function restriction of the path, since every such interval contains one with rational endpoints.

Work on this event and on the event of continuous paths. Inside any open interval

I \subset (0, \infty)

, choose two separated smaller intervals, the first to the left of the second. The first contains

r < s

with

B_{r} < B_{s}

, because its restriction is not nonincreasing. The second contains

u < v

with

B_{u} > B_{v}

, because its restriction is not nondecreasing. Thus

r < s < u < v

, all in

I

By the extreme value theorem, the path attains its maximum on

[r, v]

. This value exceeds

B_{r}

, since it is at least

B_{s}

, and exceeds

B_{v}

, since it is at least

B_{u}

. A maximizing time therefore lies in

(r, v)

and is a local maximum of Brownian motion. Every open interval in the half-line contains a positive-time interval of this kind. Consequently the set of local maxima of Brownian motion is dense in $[0, \infty)$ almost surely.