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.

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 $x$ if and only if $\sequence{y_n}$ 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 $x$ if and only if $\sequence{y_n}$ is a representative of the equivalence class $x$.

$\blacksquare$