Hensel's Lemma/P-adic Integers/Lemma 10

Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p$.

Then:
 * $\forall x \in \Z_p: p^k x \equiv 0 \pmod{p^{k+1}\Z_p} \implies x \equiv 0 \pmod{p\Z_p}$

Proof
We have: