basic properties of a limit along a filter

Let $\mathcal{F}$ be a free filter (non-principal filter) and $(x_{n})$ be a real sequence.

(i)

If $\lim\limits_{n\to\infty}x_{n}=L$ then $\operatorname{\mathcal{F}\text{-}\lim}x_{n}=L$ .
(ii)

If $\operatorname{\mathcal{F}\text{-}\lim}x_{n}$ exists, then $\liminf x_{n}\leq\operatorname{\mathcal{F}\text{-}\lim}x_{n}\leq\limsup x_{n}$ .
(iii)

The $\mathcal{F}$ -limits are unique.
(iv)

$\operatorname{\mathcal{F}\text{-}\lim}(a.x_{n}+b.y_{n})=a.\operatorname{% \mathcal{F}\text{-}\lim}x_{n}+b.\operatorname{\mathcal{F}\text{-}\lim}y_{n}$ $($ provided the $\mathcal{F}$ -limits of $(x_{n})$ and $(y_{n})$ exist $)$ .
(v)

$\operatorname{\mathcal{F}\text{-}\lim}(x_{n}.y_{n})=\operatorname{\mathcal{F}% \text{-}\lim}x_{n}.\operatorname{\mathcal{F}\text{-}\lim}y_{n}$ $($ provided the $\mathcal{F}$ -limits of $(x_{n})$ and $(y_{n})$ exist $)$ .
(vi)

For every cluster point $c$ of the sequence $x_{n}$ there exists a free filter $\mathcal{F}$ such that $\operatorname{\mathcal{F}\text{-}\lim}x_{n}=c$ . On the other hand, if $\operatorname{\mathcal{F}\text{-}\lim}x_{n}$ exists, it is a cluster point of the sequence $(x_{n})$ .

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