P-adic Expansion is a Cauchy Sequence in P-adic Norm/Converges to 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 $\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$ be a $p$-adic expansion.


Then the sequence of partial sums of the series:

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

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

Proof

From P-adic Expansion is a Cauchy Sequence in P-adic Norm, the sequence of partial sums of the series:

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

is a Cauchy sequence in the rationals $\Q$ with the $p$-adic norm.


From Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers, the sequence of partial sums of the series:

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

is a Cauchy sequence in the $p$-adic numbers $\Q_p$ with the $p$-adic norm.


By definition of the $p$-adic numbers $\Q_p$, the $p$-adic numbers $\Q_p$ form a complete normed division ring.

By definition of a complete normed division ring, every Cauchy sequence is convergent in $\struct{\Q_p, \norm{\,\cdot\,}_p}$.

$\blacksquare$

Sources