MHB How to Verify a Radical Equation Algebraically?

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Verify that both sides of the radical equation agree without using a calculator. See picture. How can this be done algebraically?

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You asked this question before:

https://mathhelpboards.com/pre-calculus-21/radical-equation-2-a-23551.html
 
It's good to know that this site keeps a record of posted questions as reference notes.
 
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I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

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