Leigh.Samphier/Sandbox/Representative of P-adic Number is Representative of Equivalence Class

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Theorem

Let $p$ be any prime number.

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


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 with $p$-adic norm.


By definition of the $p$-adic numbers as quotient of Cauchy sequences:

$\Q_p$ 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 $\eqclass{\sequence{x_n}}{}$

$\blacksquare$