I have been studying Cauchy criterion for sequences, and have come across a rather simple proof for the harmonic series, and why it diverges.
More so, we have the following:
$$\sum_{n=1}^{\infty}\frac 1n=\infty\Rightarrow divergent$$
Here is my simple proof:
Consider the sequence $\left\{a_n\right\}_{n=1}^{\infty}$ such that $a_n=\frac 1n$, then $\forall \epsilon>0,\exists \ N \in\mathbb{R}$ for $m,n\in \mathbb{N}$, such that,
$$m,n>N\Rightarrow|a_m-a_n|<\epsilon$$
Pick $m=2n$, then we have the following:
$$|a_{2n}-a_n|=\sum_{k=n+1}^{2n}\frac {1}{k}\geq \sum_{k=n+1}^{2n} \frac {1}{2n}=\frac 12$$
So pick $\epsilon=\frac 12\Rightarrow \left\{a_n\right\}_{n=1}^{\infty}$ is not Cauchy and thus divergent.
My question is, are there any other slick and easy proofs for the above claim, and if so, what are they? This series at first surprised me, as it initially doesn't seem divergent.
Thanks in advance!