Basel Problem

The next inequality was derived by Prof.Oshima.

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