P-adic Expansion is a Cauchy Sequence in P-adic Norm/Represents a P-adic Number

Theorem
Let $p$ be a prime number.

Let $\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$ be a $p$-adic expansion. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic Numbers as the Quotient of Cauchy Sequences.

Then the sequence of partial sums of the series:
 * $\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$

represents an element of $\struct {\Q_p,\norm {\,\cdot\,}_p}$.

Proof
From P-adic Expansion is a Cauchy Sequence in P-adic Norm, the sequence of partial sums of the series:
 * $\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$

is a Cauchy Sequence.

By definition, the sequence of partial sums of the series:
 * $\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$

is a representative of the equivalence class containing the sequence of partial sums.