I have $2+\sin(x)-x=0$ and I need to formulate so that it converges in $[2,3]$. I have verified that the formula $x=2+\sin(x)$ gets a convergence in this region.
Now to verify the assumptions my formulation has to satisfy in order to get convergence, I have done the following.
$2+\sin(2)-2=0.9$ and $2+\sin(3)-3=-0.85$. This ensures there is a root in this interval.
I also know that $f'(x)=cos(x)$ and $f'(2)=-0.41$ and $f'(3)=-0.99$ Since these have absolute values less than one does this ensure convergence?
No, you need to argue that $|f'(x)|<1$ for all $x$ in $[2,3]$. But that is a straightforward result from trigonometry in this case.