P-adic Integer has Unique Coherent Sequence Representative/Lemma 4

Theorem
Let $p$ be a prime number.

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

Let $a$ be an equivalence class in $\Q_p$ such that $\norm{a}_p \le 1$.

Let $\sequence{\alpha_j}$ be a coherent sequence that represents $a$.

Then:
 * $\sequence{\alpha_j}$ is the only coherent sequence that represents $a$.

Proof
Let $\sequence{\alpha'_j}$ be a coherent sequence not equal to $\sequence{\alpha_j}$.

From Leigh.Samphier/Sandbox/Representatives of same P-adic Number iff Difference is Null Sequence, it needs only to be shown that $\sequence{\alpha_j - \alpha'_j}$ is not a null sequence.

Since $\sequence{\alpha'_j} \neq \sequence{\alpha_j}$ then:
 * $\exists i_0 \in \N : \alpha'_{i_0} \neq \alpha_{i_0}$

By definition of coherent sequences:
 * $0 \le \alpha_{i_0}, \alpha'_{i_0} < p^{i_0 + 1}$

From :
 * $\alpha_{i_0} \not \equiv \alpha'_{i_0} \pmod {p^{i_0 + 1}}$