40A05-EveryBoundedSequenceHasLimitAlongAnUltrafilter.tex

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\pmtitle{every bounded sequence has limit along an ultrafilter}
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\begin{document}
\begin{THM}
Let $\F$ be an ultrafilter on $\N$ and $(x_n)$ be a real bounded
sequence. Then $\Flim x_n$ exists.
\end{THM}

\begin{proof}
Let $(x_n)$ be a bounded sequence. Choose $a_0$ and $b_0$ such
that $a_0\leq x_n \leq b_0$. Put $c_0:=\frac{a_0+b_0}2$. Then
precisely one of the sets $\{n\in\N; x_n \in \intrv{a_0}{c_0}\}$,
$\{n\in\N; x_n \in \intrv{c_0}{b_0}\}$ belongs to the filter $\F$.
(Their union is $\N$ and the filter $\F$ is an ultrafilter.) We
choose $\intrv{a_1}{b_1}$ as that subinterval from
$\intrv{a_0}{c_0}$ and $\intrv{c_0}{b_0}$ for which $C:=\{n\in\N;
x_n \in \intrv{a_1}{b_1}\}$ belongs to $\F$.

Now we again bisect the interval $\intrv{a_1}{b_1}$ by putting
$c_1=\frac{a_1+b_1}2$. Denote $A:=\{n\in\N; x_n \in
\intrv{a_1}{c_1}\}$, $B:=\{n\in\N; x_n \in \intrv{c_1}{b_1}\}$. It
holds $B\cup A\cup (\N\setminus C)=\N$. By the alternative
characterization of ultrafilters we get that one of these sets is
in $\F$. The set $\N\setminus C$ doesn't belong to $\F$, therefore
it must be one of the sets $A$ and $B$. We choose the
corresponding interval for $\intrv{a_2}{b_2}$.

By induction we obtain the monotonous sequences $(a_n)$, $(b_n)$
with the same limit $\limti n a_n = \limti n b_n :=L$ such that
for any $n\in\N$ it holds $\{n\in\N; x_n \in
\intrv{a_1}{b_1}\}\in\F$.

We claim that $\Flim x_n=L$. Indeed, for any $\ve>0$ there is
$n\in\N$ such that $\intrv{a_n}{b_n}\subseteq (L-\ve,L+\ve)$, thus
$\{n\in\N; x_n \in \intrv{a_n}{b_n}\} \subseteq A(\ve)$. The set
$\{n\in\N; x_n \in \intrv{a_1}{b_1}\}$ belongs to $\F$, hence
$A(\ve)\in\F$ as well.
\end{proof}

Note that, if we modify the definition of $\F$-limit in a such way
that we admit the values $\pm\infty$, then every sequence has
$\F$-limit along an ultrafilter $\F$. (The limit is $+\infty$ if
for each neighborhood $V$ of infinity, the set $\{n\in\N; x_n\in
V\}$ belongs to $\F$. Similarly for $-\infty$.)

\begin{thebibliography}{1}

\bibitem{agg}
M.~A. Alekseev, L.~{Yu.} Glebsky, and E.~I. Gordon, \emph{On
approximations of
  groups, group actions {and Hopf} algebras}, Journal of Mathematical Sciences
  \textbf{107} (2001), no.~5, 4305--4332.

\bibitem{balste}
B.~Balcar and P.~{\v{S}}t\v{e}p\'anek, \emph{Teorie mno\v{z}in},
Academia,
  Praha, 1986 (Czech).

\bibitem{hrjech}
K.~Hrbacek and T.~Jech, \emph{{Introduction to set theory}},
{Marcel Dekker},
  New York, 1999.

\end{thebibliography}
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