My proof that a harmonic series diverges..
One way to express your argument more formally is by considering the partial sums
$$S_{2m}=\sum_{n=1}^{2m}\frac1n=\sum_{n=1}^m\left(\frac1{2n-1}+\frac1{2n}\right)\;.$$
You can bound these partial sums as you did to obtain
$$ S_{2m}\gt\sum_{n=1}^m\left(\frac1{2n}+\frac1{2n}\right)=\sum_{n=1}^m\frac1n=S_m\;.$$
If the series were to converge, the sequences $S_{2m}$ and $S_m$ would have to converge to a common limit, so their difference would have to converge to $0$ for $m\to\infty$, whereas in fact the gap between them increases with $m$.