Formal solution needed to question that looks too easy to be true about the Gauss map

Using the itineraries of the Gauss map write the continued fraction expansion of the number $0 \leqslant \alpha \leqslant 1$ such that

$$\displaystyle \alpha = \dfrac{1}{4+\dfrac{1}{3+\alpha}}$$

I know that if $\alpha$ is irrational then the itineraries of the Gauss map give the entries of the continued fraction expansions and in my book it is quoted that if $\alpha$ is irrational then.

$$\displaystyle \alpha = \dfrac{1}{a_0+\dfrac{1}{a_1+\dots \dfrac{1}{a_n+G^{n+1}(x)}}}$$

So surely it cant be as easy as saying that $G^2(\alpha)=\alpha$ so $\alpha=[4,3,4,3,4,3,4,\dots]$

Is there a better way of writing a solution to this problem? Formal answers only please, I can see it roughly works.