Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit/P-adic Expansion
< Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit(Redirected from P-adic Expansion Converges to P-adic Number iff P-adic Expansion Represents P-adic Number)
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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.
$\blacksquare$