<s>[INST] Prove by mathematical induction. TO BE PROVED:
$$
\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\dots \frac{1}{n^2}<2
$$
Please prove this by mathematical induction only.
My approach: I already proved it through graphs but i have to prove it through mathematical induction.
Prove this equation by mathematical induction that   $\frac{1}{1} + \frac{1}{2^2} + \frac{1}{3^2} + .... + \frac{1}{n^2} < 2 $  ?
 [/INST] First, let's make the claim stronger - $\displaystyle \forall n\in\Bbb N:\ 1+\frac{1}{2^2}+\frac{1}{3^2}+\dots+\frac{1}{n^2}\le2-\frac{1}{n}<2$
Base case - $n=1$ is trivial.
Step: assume the claim holds for $k$ and prove for $k+1$ $$\underbrace{1+\frac{1}{2^2}+\dots+\frac{1}{k^2}}_{\le 2-\frac{1}{k}\text{ by induction assumption}}+\frac{1}{(k+1)^2}\le 2-\frac{1}{k}+\frac{1}{(k+1)^2}\le 2-\frac{1}{k}+\frac{1}{k(k+1)}\\=2-\frac{1}{k}+\frac{1}{k}-\frac{1}{k+1}=2-\frac{1}{k+1}<2$$
Q.E.D
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