I am currently trying to show $x\geq \sin(x)$ for $x\geq 0$. Therefore, I defined $f: \mathbb{R} \to \mathbb{R}, t\mapsto t-\sin(t)$. Then, $f'(t)\geq 0$ is true and thus $f$ is monotonically increasing. Furthermore, we reach equality, when $x=0$. 

Is that enough reasoning? And how do I show that there is only one $x$ such that $x=\sin(x)$?