A proof of irrationality of $\pi$ based on the nested radicals with roots of $2$

In this work, we prove the irrationality of $\pi$ based on the nested radicals with roots of $2$ of kind $c_k=\sqrt{2+c_{k-1}}$ and $c_0=0$. Sample computations showing how the rational approximation tends to $\pi$ with increasing the integer $k$ are presented.