It is clear that $\sup\{\sqrt2,\sqrt{2+\sqrt2},\cdots\}=2$. However, I am struggling to prove this without using limits or sequences. I know that I need to show that the set bounded above, and then we can determine that it has a supremum which I can then show is equal to $2$. How can I show that this set is bounded above in an elementary way? We have not yet covered sequences, limits, or convergence in my analysis class, nor have we covered the trigonometric functions.
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