P-adic Expansion Less Intial Zero Terms Represents Same P-adic Number

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 $\mathbf a$ be an equivalence class in $\Q_p$.

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

Let $l$ be the first index $i \ge m$ such that $d_i \ne 0$

Then the series:
 * $\displaystyle \sum_{i \mathop = l}^\infty d_i p^i$

also represents $\mathbf a$.

Proof
Let $\sequence{\alpha_n}$ be the sequence of partial sums:
 * $\forall n \in \N: \alpha _n = \sum_{i \mathop = 0}^n d_{n + m} p^{n + m}$

Let $\sequence{\beta_n}$ be the sequence of partial sums:
 * $\forall n \in \N: \beta _n = \sum_{i \mathop = 0}^n d_{n + l} p^{n + l}$

Then:

By definition of $l$:
 * $m \le l$

So:
 * $\forall n \in \N : n + l - m \ge n$

Thus $\sequence{\beta_n}$ is a subsequence of $\sequence{\alpha_n}$ by definition.

From Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence:
 * $\displaystyle \sum_{i \mathop = l}^\infty d_i p^i$ is a Cauchy Sequence in $\Q$.

From Subsequence is Equivalent to Cauchy Sequence:
 * $\displaystyle \lim_{n \mathop \to \infty} {\alpha_n - \beta_n} = 0$

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

By definition of $p$-adic numbers as a quotient of Cauchy sequences:
 * $\sequence {\alpha_n}$ and $\sequence{\beta_n}$ represent the same $p$-adic number

Since $\sequence {\alpha_n}$ represents $\mathbf a$, it follows that $\sequence {\beta_n}$ represents $\mathbf a$