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

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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 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:

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


Case 1 : $\forall n \in \N : d_n = 0$

Let:

$\forall n \in \N : d_n = 0$

Then $x = 0$.

Hence:

$\norm x_p = 0 < 1$

$\Box$


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:

\(\ds \norm x_p\) \(=\) \(\ds p^{-l}\)
\(\ds \) \(\le\) \(\ds p^0\)
\(\ds \) \(=\) \(\ds 1\)

$\blacksquare$