Representative of P-adic Number is Representative of Equivalence Class

Theorem
Let $p$ be any prime number.

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

Let $x \in \Q_p$.

Then for any sequence $\sequence{y_n}$ of the rational numbers $\Q$:
 * $\sequence{y_n}$ is a representative of the $p$-adic number $x$ $\sequence{y_n}$ is a representative of the equivalence class $x$.

Proof
By definition of the $p$-adic numbers:
 * $\Q_p$ is quotient ring

By definition of a quotient ring:
 * $\Q_p$ is a coset space

By definition of a coset space:
 * Every $p$-adic number $x$ is an equivalence class

By definitions of a representative of a $p$-adic number and a representative of an equivalence class:
 * for any sequence $\sequence{y_n}$ of the rational numbers $\Q$:
 * $\sequence{y_n}$ is a representative of the $p$-adic number $x$ $\sequence{y_n}$ is a representative of the equivalence class $x$.