I was trying to review some analysis, and came across problem 3 from page 78 of Walter Rudin's Principles of Mathematical Analysis. As part of the problem, I wanted to try to write $\sqrt{2+\sqrt{2}}$ without using the square root of a square root. In other words, I wanted to express the number in the form $q_1+q_2{q_3}^s$, where $q_1, q_2, q_3, s \in \mathbb{Q}$ (or perhaps something similar). I'm aware that this is probably a duplicate of another question, but I wasn't able to find it (I wasn't sure what to search for)... Many thanks in advance!
Edit (my work so far):
I tried expressing it as the solution to the quartic $x^4-4x^2+2=0$, but this seemed futile...
Edit 2 (the original problem):
The original problem from the text states:
If $s_1=\sqrt{2}$, and $S_{n+1}=\sqrt{2+\sqrt{s_n}} (n=1,2,3,\ldots),$ prove that $\{s_n\}$ converges, and that $s_n<2$ for $n=1,2,3,\ldots$.
The problem is from the 3rd chapter of the book, which talks about sequences and series. Rudin provides numerous theorems on this topic, such as the comparison, ratio, and root tests for convergence.