= Solution
First prove <nowhere monotonicity of Brownian motion>. On a fixed interval $[a,b]$ 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 \b[the set of <local maxima of Brownian motion> is dense in $[0,\infty)$ <almost surely>].
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