Is $a_n=\sum^n_{k=1} \frac{1}{k}$ a cauchy sequence?

Question

A cauchy sequence is bounded and $a_n=\sum^n_{k=1} \frac{1}{k}$ is unbounded. But because each term is smaller than the next it seems like I can find an $N$ such that the difference between all terms past $N$ is less than $\epsilon$ which would make it a cauchy sequence. Is it true that there is a sub sequence on $a_n$ right?

Answer 1

Never mind silly question. The terms get infinitely big so there is always a later term larger than $\epsilon$.