Prove that the harmonic series diverges

Show that
$$\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$$
does not converge.

The terms go to zero, and the partial sums are in no hurry. They first pass $10$ at around $n=12,367$, and first pass $20$ somewhere near $n=2.7\times 10^8$. Every number you can actually compute suggests the thing is settling down. It is not.

So the theorem is easy to state, easy to prove, and very hard to believe from the evidence. Post a proof — and if you can, post one where the slowness falls out of the same argument, rather than sitting beside it as an embarrassment.