Let

\begin{align*} c_n &= n!(e-\sum_{k=0}^n \frac{1}{k!}) \\ u_n &= \lfloor{\frac{1}{c_n} \rfloor} \\ v_n &= \lfloor{\frac{1}{1/c_n-\lfloor{u_n} \rfloor}} \rfloor \end{align*}

Are $u_n = n$ and $v_n = n+1$ for all $n \geq 0$?