A telescoping sum that proves ζ(2) converges

1. For $N\ge3$ the partial sums telescope with a step of $2$:
$$S_N=\frac12\sum_{n=2}^N\left(\frac1{n-1}-\frac1{n+1}\right)=\frac12\left(1+\frac12-\frac1N-\frac1{N+1}\right)\longrightarrow\frac34.$$

2. For $n\ge2$, $\frac1{n^2}<\frac1{n^2-1}$. So the partial sums of $\sum\frac1{n^2}$ are increasing and bounded by $1+\frac34=\frac74$: the series converges, with sum less than $\frac74$.

Remark. The true value is $\frac{\pi^2}6\approx1.645$, and $\frac74=1.75$: a crude bound obtained with no calculus at all.