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

Let $y \in \Q_p$.

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

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

Proof
We have:

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

From Residue Classes form Partition of Integers:
 * $\exists \mathop 0 \le i \le p^\paren {m - n} - 1: p^\paren {m - n} \divides k - i$

By definition of the $p$-adic norm: $\norm {k - i}_p \le p^\paren {n - m}$

It follows that: