Equivalence of Definitions of P-adic Integer

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

Definition 1 implies Definition 2
Let $x \in \Q_p$ such that $\norm x_p \le 1$.

From P-adic Integer is Limit of Unique P-adic Expansion, there exists a $p$-adic expansion of the form:
 * $\ds \sum_{n \mathop = 0}^\infty d_n p^n$

By definition of the canonical expansion:
 * $\ds \sum_{n \mathop = 0}^\infty d_n p^n$ is the canonical expansion of $x$

It follows that the canonical expansion of $x$ contains only positive powers of $p$.

Definition 2 implies Definition 1
Let the canonical expansion of $x$ contain only positive powers of $p$.

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

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

Then $x = 0$.

Hence:
 * $\norm x_p = 0 < 1$

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:
 * $l \ge 0$

From P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient:
 * $\norm x_p = p^{-l}$

Thus: