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

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

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

Then:
 * the canonical expansion of $x$ contains only positive powers of $p$

Proof
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 $p$.