I am new to Hensel's lemma and my syllabus uses the following theorem:
$(\mathbb Z_p $ denotes the p-adic integers)
Given $f\in \mathbb Z_p[X]$ , if $\exists \alpha_0\in \mathbb Z_p$ :$f(\alpha_0)=0$ mod $p$ but $f'(\alpha_0)\neq 0$ mod $p$, then $\exists!\alpha\in \mathbb Z_p$ :$f(\alpha)=0$ and $\alpha=\alpha_0$ mod $p$.
Here one uses the following (equivalent?) theorem: if $f\in \mathbb Z_p[X]$ and $\beta\in \mathbb Z/p\mathbb{Z}$ is a multiplicity one root of $\overline{f}$, then there is a unique lift $\beta$ to a root of $f$.
I don't understand how these two theorems say the same thing. Can we see an inclusion $\mathbb{Z}/p\mathbb{Z}\subset \mathbb{Z}_p$ or is there something deeper?
Any help would be appreciated.
For this to make sense, you need to have an understanding that $\mathbb{Z}_p$ is the inverse limit of $\mathbb{Z}/p^n\mathbb{Z}$ for $n=1, 2, \dots$. If you aren't familiar with inverse limits, then I highly recommend learning about them. They are crucial for understanding the $p$-adics, and putting them off will only hurt in the long run.
In short, a $p$-adic integer can be constructed by giving a consistent system of residues modulo $p^n$ for each $n$. This is exactly what the lifting theorem does.
Edit: Hensel's lemma is often stated as lifting a solution modulo $p^n$ to one modulo $p^{n+1}$, see for example here. Doing this for every $n$, one gets a set of solutions $(s_1, s_2, \dots)$ where $s_n$ is a solution modulo $p^n$. This is a $p$-adic integer, by definition of the $p$-adic integers as an inverse limit.