Do we have nested square roots with initial and final term and infinite terms in between? For example $$\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\cdots+\sqrt{3}}}}}$$ which happens in modified Viète nested radical
Or nested radicals like general form as follows $$\sqrt{a-\sqrt{a+\sqrt{a+\sqrt{a+\cdots+\sqrt{b}}}}}$$ If there are, what is the proof?
Thanks in advance
It has been mention in the comments but the actual value of the infinityth root does not matter, and does not really exist. To that note we may as well say: $$y=\sqrt{a-\sqrt{a+\sqrt{a+...\sqrt{b}}}}=\sqrt{a-\sqrt{a+\sqrt{a+...\sqrt{a}}}}$$ This can be easily solved by saying: $$x=\sqrt{a+\sqrt{a+...}}$$ $$x=\sqrt{a+x}\Rightarrow x^2-x-a=0$$ Which I believe has the solution: $$x=\frac{1+\sqrt{4a+1}}{2}$$ Now we can substitute this in to get: $$y=\sqrt{a-\frac{1+\sqrt{4a+1}}{2}}$$