# zero sequence

Let a field $k$ be equipped with a rank one valuation $|.|$. A sequence

$\u27e8{a}_{1},{a}_{2},\mathrm{\dots}\u27e9$ | (1) |

of elements of $k$ is called a *zero sequence* or a *null sequence*, if $\underset{n\to \mathrm{\infty}}{lim}{a}_{n}=0$
in the metric induced by $|.|$.

If $k$ together with the metric induced by its valuation^{} $|.|$ is a
complete ultrametric field, it’s clear that its sequence
(1) has a limit (in $k$) as soon as the sequence

$$\u27e8{a}_{2}-{a}_{1},{a}_{3}-{a}_{2},{a}_{4}-{a}_{3},\mathrm{\dots}\u27e9$$ |

is a zero sequence.

If $k$ is not complete^{} with respect to its valuation $|.|$, its
completion (http://planetmath.org/Completion) can be made as follows. The
Cauchy sequences^{} (1) form an integral domain^{} $D$ when the
operations “$+$” and “$\cdot $” are defined componentwise. The
subset $P$ of $D$ formed by the zero sequences is a
maximal ideal^{}, whence the quotient ring^{} $D/P$ is a field
$K$. Moreover, $k$ may be isomorphically embedded into $K$ and
the valuation $|.|$ may be uniquely extended to a valuation of
$K$. The field $K$ then is complete with respect to $|.|$ and $k$
is dense in $K$.

Title | zero sequence |
---|---|

Canonical name | ZeroSequence |

Date of creation | 2015-07-10 21:03:45 |

Last modified on | 2015-07-10 21:03:45 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 11 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 40A05 |

Synonym | null sequence |