P-adic Expansion is a Cauchy Sequence in P-adic Norm/Converges to P-adic Number

Theorem

Let $p$ be a prime number.

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

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:

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

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:

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

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:

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

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

From P-adic Numbers form Completion of Rational Numbers with P-adic Norm:

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

• 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.4$ The field of $p$-adic numbers $\Q_p$