TITLE: Non-trivial convergent sequence in Stone-Čech compactification of $\mathbb{N}$
QUESTION [8 upvotes]: Why are there only trivial convergent sequences in the Stone-Čech compactification of $\mathbb{N}$?

REPLY [8 votes]: Let me give a fairly direct proof. 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 $(\mathcal{V}_{n})_{n}$ of $(\mathcal{U}_{n})_{n}$ 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 $B_{n}=A_{n}\setminus(A_{0}\cup...\cup A_{n-1})$, then the sequence $(B_{n})_{n}$ is pairwise disjoint and $B_{n}\in\mathcal{V}_{m}$ iff $m=n$. If we let $B=\bigcup_{n}B_{2n}$, then $B\in\mathcal{V}_{m}$ if and only if $m$ is even. In other words, if $\mathcal{B}=\{\mathcal{V}\in\beta X|B\in\mathcal{V}\}$, then $\mathcal{B}$ is a clopen set with $\mathcal{V}_{m}\in\mathcal{B}$ 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.