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

Theorem
Let $p$ be a prime number.

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

Let $y \in \Q_p$

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

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

Then:
 * $\norm y_p \le p^{-n}$

Proof
We have that:

By assumption:
 * $\norm{ y - i p^n} \le p^{-m} \le p^{-n}$

and:

Hence:
 * $\max \set {\norm{ y - i p^n}_p, \norm{i p^n}_p} \le p^{-n}$

So:
 * $\norm{ y }_p \le p^{-n}$