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

Theorem
Let $p$ be a prime number.

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

Let $n, m \in Z$, such that $n < m$.

Then:
 * $\forall y \in \Q_p: \norm y_p \le p^{-n}$ there exists $i \in \Z$ such that:
 * $(1)\quad 0 \le i \le p^\paren {m - n} - 1$
 * $(2)\quad \norm {y - i p^n}_p \le p^{-m}$

Necessary Condition
Let $y \in \Q_p$.

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

Sufficient Condition
Let $y\in \Q_p$.

Let there exist $i \in \Z$ such that:
 * $(1) \quad 0 \le i \le p^\paren {m - n} - 1$
 * $(2) \quad \norm {y - i p^n}_p \le p^{-m}$