What does it mean for an ultrafilter to have a limit? I got this question from the construction of the Stone-Čech compactification using ultrafilters given in Wikipedia. There they say that if $F$ is an ultrafilter in a compact Hausdorff space $K$ then it has a unique limit.
I can see this in the case $F$ is principal. It makes sense to think about a limit of an ultrafilter $F$ as a common limit point to all subsets which are in $F$. If $F$ is principal, clearly there is only one such limit point, namely the generator.
If $F$ is not principal, then all subsets in $F$ are infinite, and by compactness of $K$ one can find subsequence for each of these subsets that has a limit point (unique by Hausdorffness). But how do I see that there is one unique limit point for a subsequence of each subset in $F$?
 A: Let $\mathscr{U}$ be an ultrafilter in the compact space $X$. Let $\mathscr{F}=\{\operatorname{cl}U:U\in\mathscr{U}\}$; then $\mathscr{F}\subseteq\mathscr{U}$, so $\mathscr{F}$ is a family of closed sets with the finite intersection property. Since $X$ is compact, $\bigcap\mathscr{F}\ne\varnothing$. Let $x\in\bigcap\mathscr{F}$; if $V$ is any open nbhd of $x$, and $U\in\mathscr{U}$, then $V\cap\operatorname{cl}U\ne\varnothing$, so $V\cap U\ne\varnothing$. Thus, $\mathscr{U}\cup\{V\}$ is a filter, and since $\mathscr{U}$ is an ultrafilter, this means that $\mathscr{U}\cup\{V\}=\mathscr{U}$, i.e., that $V\in\mathscr{U}$. Thus, every open nbhd of $x$ is in $\mathscr{U}$, which by definition means that $\mathscr{U}\to x$.
If $X$ is also Hausdorff, this limit point is unique, because $\bigcap\mathscr{F}$ is a singleton. To see this, suppose that $x,y\in\bigcap\mathscr{F}$, and $x\ne y$. If $X$ is Hausdorff, there are disjoint open sets $V$ and $W$ such that $x\in W$ and $y\in V$. But then the argument in the previous paragraph shows that $V\in\mathscr{U}$ and $W\in\mathscr{U}$, which is absurd.
A: An ultrafilter (in fact any filter) $F$ converges to $x$ if $F\supseteq (\dot{x} \cap \tau)$ where $\dot{x}$ is the one-point filter of $x$ and $\tau$ is the topology.  
The Stone Čech compactification can be constructed in many ways, for example we constructed it over an product of compact spaces, hence it was immediately clear that is compact due to Tychonoff. 
