A Proof of Irrationality of π Based on Nested Radicals with Roots of 2

In this work, we consider four theorems that can be used to prove the irrationality of $\pi$. These theorems are related to 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 tend to $\pi$ with increasing the integer $k$ are presented.