I have problem with this exercise, I really don't know how to prove it.
Let $K$ be a field that is complete with respect to a non-Archimeadean norm $ \left| \cdot \right|$. We denote by $A = \{ x \in K: \left|x \right| \leq 1 \} $ the ring of integers, $M= \{ x \in A : \left| x \right| < 1 \} $ the unique maximal ideal of $A$ and $k=A/M$ the residue field. Show that if $f \in A[T]$ is a monic polynomial and its reduction $\overline{f} \in k[T]$ has a simple root $ \overline{\alpha} \in k $, then there exists $\alpha \in A$ that is a root of $f$ and reduces to $\overline{\alpha}$, and it is a simple root of $f$ over $K$. Show also that, if $\left| \cdot \right|$ comes from a discrete valuation, then such an $\alpha $ is unique. This is a generalization of Hensel's lemma.
Hint: Consider a sequence $(a_n) \in A^{\mathbb{N}} $ satisfying $a_n = a_{n-1} - \frac{f(a_{n-1})}{f'(a_{n-1})} $ and $a_0 = \overline{\alpha}$ mod $M$. Show that $ \left| f(a_n) \right| \leq \left| f(a_0) \right|^{2^n} $. When $A$ is discretely valued with uniformizer $\omega$, uniqueness can be checked modulo $ \omega^n$ for all $n \geq 1$ as in $\mathbb{Q}_p$.
So first of all I'm not really able to prove the hint, i tried with induction but I get stuck since I don't really know how to deduce from $$ \left| f(a_n) \right| \leq \max \{ \left| a_{n-1} \right| , \left| \frac{f(a_{n-1})}{f'(a_{n-1})} \right| \} \leq \ldots \leq \left| f(a_{n-1}) \right|^2 $$ then by induction hypothesis we get the results. And okay, but once i proven this results, actually i don't see how this can help.
Maybe we have to compute $f( a_n) \mod M$ to deduce something but I really don't know. For the unicity I have no idea.
You are not inducting on a strong enough condition. See Theorem 4.1 and its proof in Section 5 here, which is based on Lang's proof of Hensel's lemma in his Algebraic Number Theory.
The uniqueness of the root in $A$ that reduces to $\overline{\alpha}$ does not require that the absolute value be discrete. See Theorem 10.5 at the link above.