Help mathematical induction Prove by mathematical induction. I was hoping if someone could give me a hint on how to solve this problem.
$$\frac{1}{1^2}+ \frac {1}{2^2} + ....+ \frac{1}{n^2} < 2 - \frac {1}{n} $$ for all integers $n\geq 2$.

*

*check this for $k=2$

*suppose it is true for $n=k$ i.e. 
$$\frac{1}{1^2}+ \frac {1}{2^2} + ....+ \frac{1}{k^2} < 2 - \frac {1}{k}$$

*Prove it for $n=k+1$ using 2


$$ \implies \frac{1}{1^2}+ \frac {1}{2^2} + ....+ \frac{1}{k^2}+\frac{1}{(k+1)^2} < 2 - \frac {1}{k}+\frac{1}{(k+1)^2}
\\ =2 -( \frac {1}{k}-\frac{1}{(k+1)^2})
\\=2 -\frac{k^2+k+1}{k(k+1)^2}
\\ < 2 -\frac{k^2+k}{k(k+1)^2}
\\= 2 -\frac{1}{k+1}$$
