A proof of irrationality of $π$ motivated by the nested radicals with roots of $2$

In this work, we prove the irrationality of $π$ motivated by 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 $π$ with increasing the integer $k$ are presented.