Hensel's Lemma/P-adic Integers/Lemma 7

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

Let $T$ be the set of $p$-adic digits.

Let $k \in \N_{>0}$.

Then:
 * $x \in \Z_p \implies \exists y \in T : y p^k \equiv x p^k \pmod {p^{k+1}\Z_p}$

Proof
Let $x \in \Z_p$.

From P-adic Integer is Limit of Unique P-adic Expansion, let:
 * $x = \ds \sum_{n \mathop = 0}^\infty d_n p^n$

We have:

Let $y = d_0$.

The result follows.