Fractional part of $1+\frac{1}{2}+\dots+\frac{1}{n}$ dense in $(0,1)$

The sequence is divergent. Let $N>0$. After the $N$-th term, the succesive terms are $<1/N$ apart. Eventually the sequence passes an integer $K$ and later $K+1$. Every point in the interval $[K,K+1]$ is then within $1/N$ from an $x_n$. So the fractional parts of $x_n$ are dense in $[0,1]$.