Can anything interesting be said about this fake proof?

$$x = \cfrac 2 {3 - \cfrac 2 {3 - \cfrac 2 {3 - \cfrac 2 {3 - \cfrac 2 {\ddots}}}}}$$

$$x = \frac 2 {3 - x}\\ x^2 - 3x + 2 = 0\\ (x-1)(x-2) = 0$$

$1,2$ are both solutions. However, if we consider this recurrence relation:

$$x_n = \frac 2 {3 - x_{n-1}}$$

when $x_{n-1} <1 \implies x_{n-1}<x_n<1$ And the squence converges to 1.

And

$$1<x_{n-1} < 2 \implies 1<x_n <x_{n-1}$$ and the sequence again converges.

but, $$2<x_{n-1} < 3 \implies x_{n-1}<x_n$$

and

$$x_{n-1} >3 \implies x_n< 0$$

The sequence isn't stable in a neighborhood of $2.$

and it converges to $1$ for nearly all starting conditions.

Simple, the problem is the assumption the three dots aka the ellipsis, in the first equation equal the ellipsis in the second. They are not.