An Elementary Proof of the Convergence of Vieta’s Product

Abstract: 

In 1593, Francois Vieta gave an expression equivalent to the infinite product,

2π=2–√2⋅2+2–√−−−−−−√2⋅2+2+2–√−−−−−−√−−−−−−−−−−−√2⋯.2π=22⋅2+22⋅2+2+22⋯.

expressing  ππ using only 2's and square roots. After more than 400 years, even today, it seems that every proof of  uses the double angle identity cos2x=2cos2x−1cos⁡2x=2cos2⁡x−1 , having the flavor of Vieta’s idea. The purpose of this note is to show that the convergence of the infinite product in can be proved without using trigonometric identities but using only the mathematics taught in high school, that is, basic algebra, Geometric Mean-Arithmetic Mean Inequality of two positive numbers and induction.