P-adic Numbers are Uncountable

Theorem
Let $p$ be any prime number.

The set of $p$-adic numbers $\Q_p$ is an uncountable set.

Proof
Let $P$ be the set of sequences on $\set{i : i \in \N : 0 \le i < p}$.

That is:
 * $P = \set{\sequence{d_n} : d_n \in \N : 0 \le d_n < p}$

From Cantor's Diagonal Argument:
 * $P$ is an uncountable set

Let $f: P \to \Q_p$ be the mapping from $P$ to $\Z_p$ defined by:
 * $\forall \sequence{d_n} \in P : \map f {\sequence{d_n}} = \ds \sum_{n = 0}^\infty d_n p^n$

where $Z_p$ denotes the $p$-adic integers and $\ds \sum_{n = 0}^\infty d_n p^n$ denotes a $p$-adic expansion

From P-adic Integer has Unique P-adic Expansion Representative:
 * $f$ is bijective

Hence:
 * $\Z_p$ is an uncountable set

Recall that $\Z_p \subseteq \Q_p$.

From Sufficient Conditions for Uncountability:
 * $\Q_p$ is an uncountable set