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

Theorem
Let $p$ be a prime number.

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

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

represents a $p$-adic number 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:
 * $\ds \sum_{n \mathop = m}^\infty d_n p^n$

is a Cauchy Sequence.

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

is a representative of a $p$-adic number by definition.