I have put up an attachment .doc file with my work. I would like to know if my proof is correct.
Thanks
It's not correct. You say that $\displaystyle \sum_{k=1}^{n}{\tfrac{1}{k^2}}=\tfrac{\pi^2}{6}$ when that is not so, actually: $\displaystyle \sum_{k=1}^{+\infty}{\tfrac{1}{k^2}}=\tfrac{\pi^2} {6}$, besides you use approximations and not the real value of the sum.
Note that: $\displaystyle \tfrac{1}{k^2}<\tfrac{1}{k\cdot (k-1)}=\tfrac{1}{k-1}-\tfrac{1}{k}$ for $\displaystyle k>1$
What do you see if you sum on both sides from $\displaystyle k=2$ to $\displaystyle k=n$ ? - add afterwards any missing terms-