I noticed this answer to the question

$\quad$ **Continued fraction of a square root**

and the comment

So I felt obliged to take this on using the theory of sequences.

Confession: I find it difficult working with continued fractions.

Let $S \gt 0$ be an integer that is not the square of another integer.

Let $a$ be the greatest positive integer satisfying $a^2 \lt S$.

Using the algebra and notation from user Julien Blanchon,

$$ x = \frac{S - a^2}{x + 2a}$$

The positive number $x$ satisfies $x \lt 1$.

Set $x_0$ to any positive number and recursively define

$$ x_{n+1} = \frac{S - a^2}{x_n + 2a}$$

Show that the sequence $a + x_n$ converges to $\sqrt S$.

It would be interesting to see if this can be explained using the theory of continued fractions.

**My Work**

I checked it out using a Python program and would bet that the claim is true. Showing it is true mathematically is another matter, and I'm hoping to see some short answers; I'm not sure how to proceed to 'deconstruct' the method and avoid an algebraic nightmare.

Also, I suspect that it doesn't really matter what you choose for the value of $a$ - any positive number will work.