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 $\eqclass{\sequence{x_n}}{}$ be a $p$-adic number of $\Q_p$.

Then any representative $\sequence{y_n}$ of the $p$-adic number $\eqclass{\sequence{x_n}}{}$ is a representative of the equivalence class $\eqclass{\sequence{x_n}}{}$.

Proof
Let $\struct{\Q, \norm {\,\cdot\,}^\Q_p}$ denote the rational numbers $\Q$ with the $p$-adic norm $\norm {\,\cdot\,}^\Q_p$.

By definition of the $p$-adic numbers:


 * $\Q_p$ is the quotient ring of Cauchy sequences of the valued field $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$

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

By definition of a coset space:
 * Every $p$-adic number $\eqclass{\sequence{x_n}}{}$ is an equivalence class

By definitions of a representative of a $p$-adic number and a representative of an equivalence class:
 * any representative $\sequence{y_n}$ of the $p$-adic number $\eqclass{\sequence{x_n}}{}$ is a representative of the equivalence class $\eqclass{\sequence{x_n}}{}$