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

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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 $p$.

$\blacksquare$