Consider the equation:
$$f(x) = \sin (x) - \ln (x) = 0$$

By Banach's contraction theorem, if we take any $x_0 \in [2.3]$ then the sequence given by:
$$x_{n+1} = x_n - \frac{\sin (x_n) - \ln (x_n)}{\cos (x_n) - \frac{1}{x_n}} \hspace{3mm} n \in \mathbb{N}$$
Converges to the root $x \in [2,3]$ with:
$$x \approx 2.21910714$$
Now, is this $x$ rational or irrational? I believe that $x$ is not only irrational but transcendental, but is it possible to prove this?