P-adic Norm not Complete on Rational Numbers/Proof 1/Case 2
Jump to navigation
Jump to search
Theorem
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for $p = 2$ or $3$.
Then:
- $\struct {\Q, \norm {\,\cdot\,}_p}$ is not a complete normed division ring.
That is, there exists a Cauchy sequence in $\struct {\Q, \norm{\,\cdot\,}_p}$ which does not converge to a limit in $\Q$.
Proof
Work In Progress You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code. |
$\blacksquare$