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.