Convergence of the series of 1/n²

For $n\ge2$ we have $n^2\ge n(n-1)$, so
$$\frac1{n^2}\le\frac1{n(n-1)}=\frac1{n-1}-\frac1n.$$

Let $S_N=\sum_{n=1}^N\frac1{n^2}$. For $N\ge2$, the sum telescopes:
$$S_N\le1+\sum_{n=2}^N\left(\frac1{n-1}-\frac1n\right)=1+1-\frac1N<2.$$

The sequence $(S_N)$ is increasing (the terms are positive) and bounded above by $2$, so it converges. Hence the series converges.