Bartle & Sherbert, edition 4, page-97 gives the following proof:
Assume that the series converges to some $S$:
$$\implies S= 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+ \dots$$
Then they proceed to derive an inequality by converting the terms of the type $\frac{1}{2n-1}+\frac{1}{2n}$ into $\frac{1}{2n}+\frac{1}{2n}=\frac{1}{n}$ $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+ \dots > \left(\frac{1}{2}+\frac{1}{2}\right)+\left(\frac{1}{4}+\frac{1}{4}\right) +\dots=1+\frac{1}{2}+\frac{1}{3} \dots=S$$ They have shown $S>S$ from the assumption, hence, there's a contradiction.
But I do not like their insistence upon using the "$\dots$", as I feel like they're hiding something behind that.
If we do a similar analysis a bit more rigorously, i.e, working solely with sequences of partial sums up to a defined, fixed $n$, (instead of those dots), we have:
$$S_{2n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4} \dots +\frac{1}{2n-1}+\frac{1}{2n}$$ $$S_{2n} >\left(\frac{1}{2}+\frac{1}{2}\right)+\left(\frac{1}{4}+\frac{1}{4}\right) \dots +\left(\frac{1}{2n}+\frac{1}{2n}\right)=1+\frac{1}{2}+\frac{1}{3} \dots \frac{1}{n}=S_n \implies S_{2n} > S_{n}$$ This is hardly news, since the 1-harmonic is monotone increasing.
Now assuming that the sequence $S_n$ converges, and passing onto the limit in the above equation yields: $$S \geq S$$ which is fine?