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.

(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 existsa 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 $(x_{n})$ .

单词	BasicPropertiesOfALimitAlongAFilter
释义	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. (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 existsa 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 $(x_{n})$ .
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