<s>[INST] A (probably trivial) induction problem: $\sum_2^nk^{-2}\lt1$ So I'm a bit stuck on the following problem I'm attempting to solve. Essentially, I'm required to prove that $\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} < 1$ for all $n$. I've been toiling with some algebraic gymnastics for a while now, but I can't seem to get the proof right. Proving it using calculus isn't a problem, but I'm struggling hither.
 [/INST] Yet another approach : 
Let us first analyze the sum till infinity. Let $$ S= \frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+ \cdots \infty$$ $$\Rightarrow S=(\frac{1}{2^2}+\frac{1}{4^2}+ \cdots\infty) +(\frac{1}{3^2}+\frac{1}{5^2}+ \cdots\infty ) $$$$\Rightarrow S= \frac{1}{4}(1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots\infty)+ S' $$ Where $$S'=\frac{1}{3^2}+\frac{1}{5^2}+ \cdots\infty $$ $$\Rightarrow S=\frac{1}{4}(1+S)+S'$$ $$\Rightarrow 3S=4S'+1........ Eqn(1)$$ Now examine the following inequality $$ (\frac{1}{2^2}-\frac{1}{3^2})+(\frac{1}{4^2}-\frac{1}{5^2})+ \cdots \infty > 0$$ $$\Rightarrow \frac{1}{2^2}+\frac{1}{4^2}+\cdots > \frac{1}{3^2}+\frac{1}{5^2}+ \cdots$$ $$\Rightarrow \frac{1}{4}(1+\frac{1}{2^2}+\cdots) >\frac{1}{3^2}+\frac{1}{5^2}+ \cdots $$ $$\Rightarrow \frac{1}{4}(1+S)> S'$$ $$\Rightarrow (1+S)> 4S'......Eqn(2)$$ From Equation 1 and 2 we get $$ 1+S> 3S-1$$ $$\Rightarrow 1> S$$ Which shows $$ 1> \frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+ \cdots \infty$$  
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