Category:P-adic Expansion is a Cauchy Sequence in P-adic Norm

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This category contains pages concerning P-adic Expansion is a Cauchy Sequence in P-adic Norm:


Let $p$ be a prime number.

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals numbers $\Q$.

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


Then the sequence of partial sums of the series:

$\ds \sum_{n \mathop = m}^\infty d_n p^n$

is a Cauchy sequence in the valued field $\struct{\Q, \norm{\,\cdot\,}_p}$.


Corollary 1

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 $\struct {\Q_p,\norm {\,\cdot\,}_p}$.


Corollary 2

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

Then the sequence of partial sums of the series:

$\ds \sum_{n \mathop = m}^\infty d_n p^n$

converges to a $p$-adic number in $\struct{\Q_p, \norm{\,\cdot\,}_p}$.