Category:Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit

This category contains pages concerning Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit:

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 $\sequence{x_n}$ be a rational sequence.

Then:

$\sequence{x_n}$ converges to $a$ if and only if $\sequence{x_n}$ is a representative of $a$

Corollary

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 $a$