The Cauchy criterion

1. For $m>n\ge1$,
$$|u_m-u_n|=\sum_{k=n+1}^m\frac1{k^2}\le\sum_{k=n+1}^m\left(\frac1{k-1}-\frac1k\right)=\frac1n-\frac1m<\frac1n.$$
Given $\varepsilon$, take $N>\frac1\varepsilon$: for $m>n\ge N$, $|u_m-u_n|<\varepsilon$. The sequence is Cauchy, hence converges.

2. $H_{2n}-H_n=\sum_{k=n+1}^{2n}\frac1k\ge n\cdot\frac1{2n}=\frac12$. So no matter how large $N$ is, taking $n\ge N$ and $m=2n$ gives $|H_m-H_n|\ge\frac12$: the sequence is not Cauchy, hence diverges.

Remark. The Cauchy criterion is powerful because it proves convergence *without knowing the limit*. (Here the limit of $u_n$ is $\frac{\pi^2}6$, which is much harder to find.) Note that in part 2, consecutive terms $H_{n+1}-H_n=\frac1{n+1}\to0$, yet the sequence diverges: the criterion needs *all* pairs $m,n$.