Equivalence of Definitions of P-adic Integer/Definition 2 Implies Definition 1

Theorem
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the valued field of $p$-adic numbers for some prime $p$.

That is, such that:
 * $\Q_p$ is the field of $p$-adic numbers
 * $\norm {\,\cdot\,}_p$ is the $p$-adic norm on $\Q_p$.

Let $x \in \Q_p$ such that the canonical expansion of $x$ contains only positive powers of $p$.

Then:
 * $\norm x_p \le 1$

Proof
Let the canonical expansion of $x$ contain only positive powers of $p$.

That is:
 * $x = \ds \sum_{n \mathop = 0}^\infty d_n p^n : \forall n \in \N : 0 \le d_n < p$

Case 1 : $\forall n \in \N : d_n = 0$
Let:
 * $\forall n \in \N : d_n = 0$

Then:

Hence:

Case 2 : $\exists n \in \N : d_n > 0$
Let:
 * $\exists n \in \N : d_n > 0$

Let:
 * $l = \min \set {i: i \ge 0 \land d_i \ne 0}$

Hence:
 * $l \ge 0$

Thus: