Is the sequence $(x_n)_{n=1}^\infty$, where $x_n$ is the fractional part of $1+\frac{1}{2}+\dots+\frac1n$, dense in $(0,1)$? The fractional part of a number $y$ is defined as $y-\lfloor y\rfloor$.
For a sequence like $a,2a,3a,\dots$ where $a$ is an irrational number, it is known that the fractional part sequence is dense. (I think there's even a name for this result, but I can't recall.) The proof uses a pigeonhole-style argument to show that the sequence must fall into any small interval of $(0,1)$ and relies on the linearity of the sequence, which we don't have in our sequence.