I'm trying to prove that the function $f$ is of class $C^\infty$ as follows. Given any integer $n\geq 0$, define $f_n : \mathbb R \rightarrow \mathbb R$ by
$f_n(x) = \frac{e^{-1/x}}{x^n}, \quad \text{for} \quad x > 0,$
$f_n(x) = 0, \quad \text{ for } \quad x<=0.$
Question: How do show that $f_n$ is continuous at 0, differentiable at 0, and f'_n(x) = f_{n+2}(x) -nf_{n+1}(x) for all $x$, and $f_n$ is of class $C^\infty$?