P-adic Norm of p-adic Number is Power of p/Lemma
Theorem
Let $p$ be a prime number.
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.
Let $\sequence {x_n}$ be a Cauchy sequence in $\struct{\Q, \norm {\,\cdot\,}_p}$ such that $\sequence {x_n}$ does not converge to $0$.
Then:
- $\exists v \in \Z: \ds \lim_{n \mathop \to \infty} \norm{x_n}_p = p^{-v}$
Proof
From P-adic Norm on Rational Numbers is Non-Archimedean Norm, $p$-adic norm is a non-Archimedean norm on the rationals $\Q$.
Since $\sequence {x_n}$ does not converge to $0$, from Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary:
- $\exists N \in \N: \forall n, m > N: \norm {x_n}_p = \norm {x_m}_p$
By definition of the $p$-adic norm on $\Q$:
- $\exists v \in \Z: \norm {x_{N + 1} }_p = p^{-v}$
Therefore:
- $\forall n > N: \norm {x_n}_p = \norm {x_{N + 1} }_p = p^{-v}$
Let $\sequence {y_n}$ be the subsequence of $\sequence {\norm {x_n}}$ defined by:
- $\forall n: y_n = \norm {x_{N + n} }_p$
Then $\sequence {y_n}$ is the constant sequence $\tuple {p^{-v}, p^{-v}, p^{-v}, \dotsc}$.
Therefore:
\(\ds \lim_{n \mathop \to \infty} \norm {x_n}_p\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} y_n\) | Limit of Subsequence equals Limit of Real Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds p^{-v}\) | Constant Sequence Converges to Constant in Normed Division Ring |
$\blacksquare$