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

Jump to navigation
Jump to search

This article needs proofreading.Please check it for mathematical errors.If you believe there are none, please remove `{{Proofread}}` from the code.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Proofread}}` from the code. |

## 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 $p$.

$\blacksquare$