Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit/P-adic Expansion

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Theorem

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$.

Let $\ds \sum_{n \mathop = m}^\infty d_n p^n$ be a $p$-adic expansion.


Then:

$\ds \sum_{n \mathop = m}^\infty d_n p^n$ converges to $a$ if and only if $\ds \sum_{n \mathop = m}^\infty d_n p^n$ is a representative of $a$


Proof

By definition of a $p$-adic expansion:

$\ds \sum_{n \mathop = m}^\infty d_n p^n$ is a rational sequence.

The theorem follows from Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit.

$\blacksquare$