P-adic Expansion of P-adic Unit

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

Let $\Z_p$ be the $p$-adic integers.

Let $a \in \Z_p$.

Let $\ldots a_n \ldots a_3 a_2 a_1 a_0$ be the canonical expansion of $a$.

Then:
 * $a$ is a $p$-adic unit $a_0 \ne 0$

Proof
From P-adic Unit has Norm Equal to One:
 * a is a $p$-adic unit $\norm a_p = 1 = p^0$

By definition of the canonical expansion:
 * $a$ is the limit of the $p$-adic expansion $\ds \sum_{n \mathop = 0}^\infty a_n p^n$

From P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient:
 * $\norm a_p = p^0$ $a_0 \ne 0$

The result follows.