I'm trying to find a proof for the following formulation of Hensel's lemma:
$$\text{Let } f \in \mathbb{Z}[x_1, \dots, x_n], a = (a_1, \dots, a_n) \text{ be such that (with } p \text{ prime)}$$
$$ f(a) = 0 \pmod{p} $$
$$ \text{but } \frac{\partial f}{\partial x_i}(a) \neq 0 \pmod{p} \text{ for some } 0 \leq i \leq n $$
$$ \text{then for all } m\in\mathbb{N} \text{ there exists } b = (b_1, \dots, b_n) \text{ such that}$$
$$ f(b) = 0 \pmod{p^m} \text{ and } b_i \equiv a_i \pmod{p} \text{ for } i = 1, \dots, n$$
Well, to be honest, I have no idea where to begin with. I was thinking about an induction proof, but I haven't got a clue how to construct such solutions.
I would appreciate some help