# basic properties of a limit along a filter

###### Theorem 1.

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

1. (i)

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

2. (ii)

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

3. (iii)

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

4. (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$)$.

5. (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$)$.

6. (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 BasicPropertiesOfALimitAlongAFilter 2013-03-22 15:32:23 2013-03-22 15:32:23 kompik (10588) kompik (10588) 9 kompik (10588) Theorem msc 03E99 msc 40A05