What is the limit of this sequence? Problem 3 in the Exercises after Chapter 3 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: 
Let $s_1 \colon= \sqrt{2}$, and let 
$$s_{n+1} \colon= \sqrt{2+\sqrt{s_n}} \mbox{ for } n = 1, 2, 3, \ldots. $$ 
Then how to rigorously calculate $$\lim_{n\to\infty} s_n,$$ justifying each step from the definition of convergence of a sequence and the theorems on convergence of sequences as have been proved by Rudin upto this point in his book?  
I know that this sequence is increasing and bounded (above) and hence convergent. 

Continuing Cheung SW's answer and solving the cubic, we get $$k=\frac{1}{3} \left(1-2 \sqrt[3]{\frac{2}{47+3 \sqrt{249}}}+\sqrt[3]{\frac{1}{2}
   \left(47+3 \sqrt{249}\right)}\right)\approx 1.35321$$ and then $$L=\frac{1}{3} \left(-1+\sqrt[3]{\frac{79}{2}-\frac{3
   \sqrt{249}}{2}}+\sqrt[3]{\frac{1}{2} \left(79+3 \sqrt{249}\right)}\right)\approx 1.83118$$
