Q:

Show that for every integer $n ≥ 1$, $1 + \frac{1}{4} +\frac{1}{9} + · · · + \frac{1}{n^2} ≤ 2 − \frac{1}{n}$

I can just think of trying to prove $\frac{1}{4} +\frac{1}{9} + · · · + \frac{1}{n^2} ≤ \frac{1}{n}$, but remains stuck.

A:

Hint: Note that 
$$\frac{1}{2^2}+\frac{1}{3^2}+\cdots +\frac{1}{n^2} \lt \frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\cdots+\frac{1}{(n-1)\cdot n}.$$
The expression on the right turns out to be a telescoping sum. For $\frac{1}{(i-1)\cdot i}=\frac{1}{i-1}-\frac{1}{i}$. 
One could also do an induction version of the above proof, using the fact that $\frac{1}{(n+1)^2}\lt \frac{1}{n}-\frac{1}{n+1}$. 

