Prove that sqrt(1+sqrt(1+sqrt(1+sqrt(...)))) = the golden ratio (ILIEKMATHPHYSICS)

This video references the book "Introduction to Real Analysis" by Bartle and Sherbert (Fourth Edition). For more details, see section 3.3 on the Monotone Convergence Theorem.
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Real Analysis (Bartle and Sherbert) Playlist: • Real Analysis (Bartle)...

[@stenzenneznets]

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[@vlynn5695]

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[@chaosredefined3834]

To prove the fact that it's increasing, use proof by induction...
Base case: x1 = 1, x2 = sqrt(2), so x2 > x1. Easy base case.
Inductive case: Assume x(n) > x(n-1). Then 1 + x(n) > 1 + x(n-1). Therefore, sqrt(1 + x(n)) > sqrt(1 + x(n-1)). Therefore x(n+1) > x(n). Inductive case done.

[@lorenzosaudito]

Nice