So, today in my real analysis course, a theorem was given:
$\sum a_n \text{ converges }\iff \{s_n\}_{n=1}^{\infty} \text{ is bounded,}$
where $s_n$ is the $n$-th partial sum of $\sum a_n.$ Now, of course this theorem makes perfect sense to me, but I began to wonder if there is a way to use this and only this to prove the divergence of the harmonic series. So I have been led to two questions...
a) Given $n \in \mathbb{N},$ is there a way to find the $m$-th harmonic number closest (either above or below) $n$?
b) Given $M \in \mathbb{R},$ how can one show that there exists $N \in \mathbb{N}$ so that $\sum_{n=1}^N \frac{1}{n} > M,$ with only the above theorem and perhaps theorems primitive to it?
Thanks a lot! This isn't a homework problem, just a consideration of mine before falling asleep -- I can't exactly try to work on it myself with class in 7 hours ):