Given a difference equation $a_{n+k}=f(a_n,a_{n+1},\dots,a_{n+k-1})$, we can classify $n=\infty$ as an ordinary, regular singular, or irregular singular point by classifying $x= \infty$ in the differential equation analogue.
If $n=\infty$ is ordinary, or regular singular, then the solution to the difference equation is asymptotic to the solution to the differential equation as n grows large (Bender & Orzag).
$$ a_n \sim y(n) \quad n \rightarrow \infty \>. $$
Does anyone know where I can find a proof of this? The book I am reading (Bender & Orzag) just states this as fact without proof. I would really like to be shown why this is so.