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

Let $y \in \Q_p$

Let $n \in Z$.

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

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

Proof
Now:

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

and:

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

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