Question about Stone-Čech compactification Assume $R=(\mathcal{U}_n)_n$ is a sequence of distinct ultrafilters on some set $X$. Since every Hausdorff space has an infinite discrete subspace, there is a subsequence $R=(\mathcal{V}_n)_n$ of $R=(\mathcal{U}_n)_n$ such that $\{\mathcal{V}_n|n \in \mathbb{N} \}$ is a discrete subspace of $\beta X$. In particular, there is a sequence $(A_n)_n$ of sets with $A_n \in \mathcal{V}_m$ if and only if $m =n$. If we let $B_n=A_n - (A_0 \cup...\cup A_{n−1})$, then the sequence $(B_n)_n$ is pairwise disjoint and $B_n \in \mathcal{V}_m$  iff $m=n$. If we let $B =\cup_n B_{2n}$, then $B \in \mathcal{V}_m$ if and only if $m$ is even. In other words, if $B =\{ \mathcal{V} \in βX|B \in \mathcal{V}\}$, then $B$ is a clopen set with $\mathcal{V}_m \in B$ iff $m$ is even. Therefore, we conclude that the sequence $(\mathcal{V}_m)_m$ cannot converge to any point, so the sequence $(\mathcal{U}_n)_n$ cannot converge to any point either.

I would like to know that:
(1) Why "$B_n \in \mathcal{V}_m$  iff $m=n$"? And " $B =\cup_n B_{2n}$, then $B \in \mathcal{V}_m$ if and only if $m$ is even "?
(2)$B$ is a clopen set with $\mathcal{V}_m \in B$ iff $m$ is even

 A: (1a) If $m=n$ then $\mathcal V_m$ contains $A_n$ but contains none of $A_0,A_1,\dots,A_{n-1}$. Therefore, being an ultrafilter, it doesn't contain $A_0\cup\dots\cup A_{n-1}$ and does contain $A_n-(A_0\cup\dots\cup A_{n-1})=B_n$.  On the other hand, if $m\neq n$, then $\mathcal V_m$ doesn't contain $A_n$ and therefore cannot contain any subset of $A_n$, such as $B_n$.
(1b) If $m$ is even, say $m=2k$, then, by (1a), $\mathcal V_m$ contains $B_{2k}$ and therefore also its superset $B$.  On the other hand, if $m$ is odd, then $\mathcal V_m$ contains $B_m$ which is disjoint from all $B_{2n}$ and therefore from their union $B$.  So $\mathcal V_m$ cannot contain $B$.
(2) Some notation got lost when you transcribed the question. $\{\mathcal V\in\beta X:B\in \mathcal V\}$ is certainly not equal to $B$, but it is the closure $\overline B$ of $B$ in $\beta X$.  It is clopen because its complement is also a closure, namely $\overline{X-B}$.  And we have $\mathcal V_m\in\overline B$, which means by definition $B\in\mathcal V_m$, if and only if $m$ is even, by (1b).
