Banach's Fixed point aplication

In analysis, a usual sufficient condition for the convergence of a iteration $x_n=g(x_{n-1})$ is that $g$ be continuously differentiable and $$|g'(x)|\leq \alpha <1$$

Verify this by the use of the Banach's fixed point theorem.

Try:

As $g$ is continuously differentiable by the Mean value theorem we get

$$|g(x)-g(y)|=|g'(c)||x-y|\leq \alpha |x-y|$$

then $g$ is a contraction, and therefore by the Banach fixed point theorem $g$ has precisely one fixed point.

Is everything okay, or how could I improve it?

It is correct, but I would underline a very important property: you want to work in a complete metric space, otherwise you can have a contraction mapping and the theorem may fail.

If you remark that $g$ is defined from $\mathbb{R}$ to $\mathbb{R}$ and that $\mathbb{R}$ with the euclidean metric is complete, then it is totally fine.

Historical note: sometimes the theorem is called Banach-Caccioppoli fixed point theorem, since they both formulated it independently around 1922 and 1931 respectively.