<s>[INST] Is there any way to prove $ \frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2} \leq \frac{\pi^2}{6} $ by induction since $ \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} $ we have that for each $n\in \Bbb N$ , $  \frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2} \leq \frac{\pi^2}{6} $
my problem is can we prove the statement 
"for each $n\in \Bbb N$ , $  \frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2} \leq \frac{\pi^2}{6} $"
only using induction ?(without using the fact that convergence of the above series)
any ideas, thanks!
 [/INST] This is unlikely. The bare induction step would be
$$S_n\le\frac{\pi^2}6\implies S_n+\frac1{n^2}\le\frac{\pi^2}6,$$ which obviously doesn't hold. There is not enough information in the inductive hypothesis.
Any information on the asymptotics of the series will be of the form $\dfrac{\pi^2}6-\epsilon(n)$, which contains "the answer" (i.e. a hint on convergence).
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