P-adic Number has Unique P-adic Expansion Representative

Theorem
Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences.

Let $a$ be an equivalence class in $\Q_p$.

Then $a$ has exactly one representative that is a $p$-adic expansion.

Case 1
Let $\norm{a}_p \le 1$.

From Leigh.Samphier/Sandbox/Equivalence Class in P-adic Integers Contains Unique P-adic Expansion $a$ has exactly one representative that is a $p$-adic expansion of the form:
 * $\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$

Case 2
Let $\norm{a}_p > 1$.