TITLE: Proving sum of $1/n^2$ is less than or equal to $2$
QUESTION [5 upvotes]: So I'm suppose to prove that $\sum 1/n^2 \le 2$. Should I use induction?

REPLY [3 votes]: Should I use induction?

Not necessarily. Note that $\dfrac1{n^2}\lt\dfrac1{n(n-1)}=\dfrac1{n-1}-\dfrac1n$ for each $n\geqslant2$ hence, telescoping the terms of the series for $n\geqslant2$ shows that
$$
\sum_{n=2}^{\infty}\frac1{n^2}\lt\sum_{n=\color{red}{\mathbf2}}^{\infty}\left(\dfrac1{n-1}-\dfrac1n\right)=\dfrac1{\color{red}{\mathbf2}-1}=1.
$$
To conclude, add the $n=1$ term $\dfrac1{1^2}=1$.