How can I show $F$ is not monotonic in any subinterval? I'm trying to solve Problem 1(b) of [Fourier Analysis] by Stein. The problem is as follows:
Choose a countable dense sequence ${r_n}$ in $[0,1]$. Consider
$\displaystyle{F(x)=\sum_{n=1}^\infty3^{-n}g(x-r_n)}$,
where $g(x)=\sin(1/x)$ when $x\neq0$, and $g(0)=0$. Then $F$ fails to be monotonic in any subinterval of $[0,1]$.
The Author gives as a hint $3^{-k}>\sum_{n>k}3^{-n}$. But I cannot figure out how to use this hint. Since $F$ is a superposition of highly oscillating functions, it seems that $F$ is not monotonic.

Let a small subinterval $I$ of $[0,1]$ be given, and let $k$ be the least number such that $r_k$ is in that interval. Now, $f_k(x)=\sum_{n=1}^{k-1}3^{-n}g(x-r_n)$ is continuous on $I$, so there exists a subinterval $[r_k,r_k+\varepsilon]$ of $I$ such that $f_k(x)$ varies by at most $3^{-k}/100$ on that interval. The function $3^{-k}g(x-r_k)$ however varies by $3^{-k}$ infinitely many times on this interval, and the remainder $\sum_{n=k+1}^\infty 3^{-n}g(x-r_n)$, by the hint in the book, is too small to destroy these variations. It follows that the full sum $\sum_1^\infty$ is non-monotonic on the interval $[r_k,r_k+\varepsilon]\subseteq I$.
