Convergence of the numerical scheme

Can easily show convergence: $$e^n\rightarrow 0 \hbox{ as }\Delta t\rightarrow 0,$$ because $$\lim_{\Delta t\rightarrow 0} \tilde\omega = \lim_{\Delta t\rightarrow 0} \frac{2}{\Delta t}\sin^{-1}\left(\frac{\omega\Delta t}{2}\right) = \omega,$$ by L'Hopital's rule or simply asking sympy: or WolframAlpha:

>>> import sympy as sp
>>> dt, w = sp.symbols('x w')
>>> sp.limit((2/dt)*sp.asin(w*dt/2), dt, 0, dir='+')
w