Representatives of same P-adic Number iff Difference is Null Sequence

Theorem
Let $p$ be a prime number.

Let $\norm{\,\cdot\,}_p$ be the $p$-adic norm on the rational numbers $\Q$.

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

Let $\sequence{\alpha_n}$ and $\sequence{\beta_n}$ be Cauchy sequences in $\struct{Q, \norm{\,\cdot\,}_p}$.

Then:
 * $\sequence{\alpha_n}$ and $\sequence{\beta_n}$ are representatives of the same equivalence class in $\Q_p$


 * the sequence $\sequence{\alpha_n - \beta_n}$ is a null sequence.
 * the sequence $\sequence{\alpha_n - \beta_n}$ is a null sequence.

Proof
Let $\mathcal C$ be the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\mathcal N$ be the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.

Then $\Q_p$ is the quotient ring $\mathcal C \, \big / \mathcal N$ by definition of the $p$-adic numbers.

By definition of the quotient ring $\mathcal C \, \big / \mathcal N$:
 * $\sequence{\alpha_n}$ and $\sequence{\beta_n}$ are representatives of the same equivalence class in $\mathcal C \, \big / \mathcal N$


 * $\sequence{\alpha_n}$ and $\sequence{\beta_n}$ are elements of the same left coset of $\mathcal N$ in $\mathcal C$.
 * $\sequence{\alpha_n}$ and $\sequence{\beta_n}$ are elements of the same left coset of $\mathcal N$ in $\mathcal C$.

From Element in Left Coset iff Product with Inverse in Subgroup:
 * $\sequence{\alpha_n}$ and $\sequence{\beta_n}$ are elements of the same left coset of $\mathcal N$ in $\mathcal C$


 * $\sequence{\alpha_n} - \sequence{\beta_n} = \sequence{\alpha_n - \beta_n} \in \mathcal N$.
 * $\sequence{\alpha_n} - \sequence{\beta_n} = \sequence{\alpha_n - \beta_n} \in \mathcal N$.

The result follows.