This question is part of a wider conjecture I have formed with someone which has its roots in Raayoni *et al.* (2019). The subfactorial function can be written as $$!n=\frac{n!}e+\frac{(-1)^n}{n+2-\dfrac1{n+3-\dfrac2{n+4-\cdots}}}$$ which is equation (17) of the MathWorld documentation of subfactorial.

Is there a proof of this identity that can be found in the literature or elsewhere?