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 $x$.
$\blacksquare$