TITLE: Can we prove the irrationality of pi from its expression as a limiting product with the nested square roots?
QUESTION [7 upvotes]: Is it possible to prove the irrationality of $ \pi $ from the expression $ \pi = Lim_{m \to \infty} [2^m(\sqrt(2-\sqrt(2+\sqrt(2+...)...)))]$, where there are $m$ square roots in the nested expression? 
This expression is obtained by considering the ratio of the perimeter of a regular $n$-sided polygon to the diameter of its circumcircle and then letting $ n \to \infty$. This expression was found by Archimedes I think.But didn't see Archimedes' name as one of those proving the irrationality. Makes me suspect whether this can be done. Can anyone please provide me a method (if it exists)? Any indirect method starting from here would be fine.

REPLY [5 votes]: No we can't. From your infinite nested radical expression for $\pi$ you cannot conclude if it is rational or irrational because infinite nested radical can take rational values. Here is an example.
$$
2 = \sqrt {2+\sqrt{2+\sqrt{2+...}}}
$$