Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Necessary Condition

Theorem
Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\Z_p$ be the $p$-adic integers.

Let $n \in Z$.

Let $y \in \Q_p$.

Let $\norm{y}_p \le p^{-n}$.

Then there exists $i \in \Z$:
 * $(1)\quad0 \le i \le p-1$
 * $(2)\quad\norm{y - i p^n}_p < p^{-n}$

Proof
Now:

From Integers are Dense in Unit Ball of P-adic Numbers:
 * $\exists \mathop m \in \Z : \norm{p^{-n} y - m}_p < 1$

From Residue Classes form Partition of Integers:
 * $\exists \mathop 0 \le i \le p-1 : p \divides m - i$

By definition of the $p$-adic norm: $\norm{m - i}_p \le p^{-1} < 1$

It follows that: