Prove that $s_n\leq 2-\frac{1}{n}$ for all $n\geq 1$ We have the sequence $(s_n)_{n\geq 1}$ given by $s_n=\sum^n_{k=1}\frac{1}{k^2}$. Prove that  $s_n\leq 2-\frac{1}{n}$ for all $n\geq 1$. Thanks in advance!
Hint

$$\frac1{k^2}\le\frac1{k(k-1)}=\frac1{k-1}-\frac1k\quad \forall k\ge2$$
and telescope.
Look at the Riemann zeta function.

$\zeta(2)=\pi^2 / 6$ which is smaller than $1.7$.

Since every value of your function is smaller than $\zeta(2)$ your claim follows.

See : http://en.wikipedia.org/wiki/Riemann_zeta_function#Specific_values