basic properties of a limit along a filter

Theorem 1.

Let F be a free filter (non-principal filter) and (xn) be a real sequence.

  1. (i)

    If limnxn=L then -limxn=L.

  2. (ii)

    If -limxn exists, then lim infxn-limxnlim supxn.

  3. (iii)

    The -limits are unique.

  4. (iv)

    -lim(a.xn+b.yn)=a.-limxn+b.-limyn (provided the -limits of (xn) and (yn) exist).

  5. (v)

    -lim(xn.yn)=-limxn.-limyn (provided the -limits of (xn) and (yn) exist).

  6. (vi)

    For every cluster pointPlanetmathPlanetmath c of the sequence xn there exists a free filter such that -limxn=c. On the other hand, if -limxn exists, it is a cluster point of the sequence (xn).

Title basic properties of a limit along a filter
Canonical name BasicPropertiesOfALimitAlongAFilter
Date of creation 2013-03-22 15:32:23
Last modified on 2013-03-22 15:32:23
Owner kompik (10588)
Last modified by kompik (10588)
Numerical id 9
Author kompik (10588)
Entry type Theorem
Classification msc 03E99
Classification msc 40A05