Proof that $\sum_{1}^{\infty} \frac{1}{n^2} <2$
Hint:
$$\sum_{n=1}^{\infty} \frac{1}{n^2} < 1+ \int_{1}^{\infty} \frac{1}{x^2}dx $$
Hint: for $n \geq 2$, $$ \frac 1{n^2} \leq \frac{1}{n(n-1)} = \frac1{n-1} - \frac 1n $$
Proof that $\sum_{1}^{\infty} \frac{1}{n^2} <2$
Hint:
$$\sum_{n=1}^{\infty} \frac{1}{n^2} < 1+ \int_{1}^{\infty} \frac{1}{x^2}dx $$
Hint: for $n \geq 2$, $$ \frac 1{n^2} \leq \frac{1}{n(n-1)} = \frac1{n-1} - \frac 1n $$