Basel Problem

 $\displaystyle\pi\theta=\sum_{k=1}^n\frac{1}{k^2}<\sum_{k=1}^m\frac{1}{k^2}+\sum_{k=m+1}^{n}\dfrac{1}{k(k-1)}=\sum_{k=1}^m\frac{1}{k^2}+\sum_{k=m+1}^{n}\left(\dfrac{1}{k-1}-\dfrac{1}{k}\right)<\sum_{k=1}^m\frac{1}{k^2}+\frac{1}{m}$