<s>[INST] Summation inductional proof: $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$ Having the following inequality
$$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$$
To prove it for all natural numbers is it enough to show that:
$\frac{1}{(n+1)^2}-\frac{1}{n^2} <2$  or  $\frac{1}{(n+1)^2}<2-\frac{1}{n^2} $
 [/INST] Here is one way...
$$\frac1{k^2} < \frac1{k(k-1)}= \frac1{k-1} - \frac1{k}$$
Now telescope to get
$$1+\sum_{k=2}^n\frac1{k^2} < 1+1-\frac1{n}< 2$$
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