Proving by induction that $\sum\limits_{k=2}^n \frac1{k^2}\le 1$

Question

I'm having trouble proving this by induction. We need to show that $P(k+1)$ is true: $$\sum_{i=2}^{k+1} \frac{1}{i^2}\leq 1.$$

Don't know where to go from here. Any help?

Answer 1

Hint. Try to apply induction to $$ \sum_{k=2}^n \frac1{k^2}\le 1-\frac1{n},\qquad n\ge2. $$

Answer 2

Note: NOT a proof by induction as demanded

Let,the whole big square represent $1$ and the smaller parts inside the terms of the series $\displaystyle\frac{1}{2^2},\frac{1}{3^2},\frac{1}{4^2},...$

Special thanks to @Alex.Jordan and @ArtOfMathematics