P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient

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 $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ be a $p$-adic expansion.

Let $\mathbf a$ be the equivalence class in $\Q_p$ containing $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$.

Let $l$ be the index of the first nonzero coefficient in the $p$-adic expansion.

That is, $l$ is the first index in the ordered set $\set{i : i \ge m \land d_i \neq 0}$.

Then:
 * $\norm{\mathbf a}_p = p^{-l}$

Proof
For all $n \ge m$, let:
 * $\alpha_n = \displaystyle \sum_{i \mathop = m}^n d_i p^i$

By assumption:
 * $\sequence{\alpha_n}$ is a representative of $\mathbf a$

By definition of the induced norm:
 * $\norm{\mathbf a}_p = \displaystyle \lim_{n \mathop \to \infty} \norm{\alpha_n}_p$

From Eventually Constant Sequence Converges to Constant it is sufficient to show:
 * $\forall n \ge l + 1 : \norm{\alpha_n}_p = p^{-l}$

Let $n \ge l + 1$.