Proving $\frac{1}{n^2}$ infinite series converges without integral test

Hint: $$\frac{1}{n^2} < \frac{1}{n(n-1)}.$$

$$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \ldots + \frac{1}{n(n - 1)} = 1 - \frac12 + \frac12 - \frac13 + \ldots + \frac{1}{n - 1}-\frac1n = 1 - \frac1n \to 1.$$

With fewer words. Hopefully clear enough. Oresme's style, but converging this time, and proving that the sum is $<2$.