Proof that π is irrational no relation y
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    {\displaystyle {\begin{aligned}g_{n+2}&={\frac {c^{n+2}}{x^{2}k(k+1)\cdots (k+n-1)}}\cdot {\frac {x^{2}}{(k+n)(k+n+1)}}f_{k+n+2}(x)\\[5pt]&={\frac {c^{n+2}}{x^{2}k(k+1)\cdots (k+n-1)}}f_{k+n+1}(x)-{\frac {c^{n+2}}{x^{2}k(k+1)\cdots (k+n-1)}}f_{k+n}(x)\\[5pt]&={\frac {c(k+n)}{x^{2}}}g_{n+1}-{\frac {c^{2}}{x^{2}}}g_{n}\\[5pt]&=\left({\frac {ck}{x^{2}}}+{\frac {c}{x^{2}}}n\right)g_{n+1}-{\frac {c^{2}}{x^{2}}}g_{n},\end{aligned}}}
  which is a linear combination of gn + 1 and gn with integer coefficients. Therefore, each gn is an integer multiple of y. Besides, it follows from claim 2 that each gn is greater than 0 (and therefore that gn ≥ |y|) if n is large enough and that the sequence of all gn converges to 0. But a sequence of numbers greater than or equal to |y| cannot converge to 0.
Since f1/2(π/4) = cos(π/2) = 0, it follows from claim 3 that π2/16 is irrational and therefore that π is irrational.
On the other hand, since
no relation