I'm a complete amateur with respect to mathematics, but I looked up a few proofs of the irrationality of $\pi$ and was unsatisfied by the lack of proofs that would be elementary enough to be able to teach using only mathematics taught at my country's equivalent of high-school (where calculus isn't taught, for instance), which is what I ultimately want to do.
In trying to come up with one such proof myself, I traced my thought back to Archimedes' approximation of $\pi$ and its respectively given lower and upper bounds of $n_k \cdot \sin (\theta_k)$ and $n_k \cdot \tan (\theta_k)$, where $n_k = 3 \cdot 2^k$ and $\theta_k = \frac{\pi}{n_k}$ for all $k \in \mathbb N$. In particular, I know that the nested interval theorem guarantees that $\pi$ is the limit of the sequence $([A_k, B_k])_{k \in \mathbb N}$ of nested intervals, and this theorem is fairly intuitive enough, and indeed axiomatic with respect to the real numbers, to justify invoking it in such an elementary proof.
My question is: is it possible to prove the irrationality of $\pi$, defining it as this limit of nested intervals, from the nested interval theorem? As in, similarly to how certain irrational numbers, like $\sqrt 2$, can be proven to be so by their definition as limits of a limiting process related to a completeness axiom and/or construction of the real numbers; and if so, how should I go about proving the irrationality of $\pi$ in this way?