= No nontrivial convergent sequences in the Stone-Čech compactification
Every <convergent sequence> in the <Stone-Čech compactification of the natural numbers> is eventually constant. If distinct <ultrafilters> $U_n$ converged, isolate each from the other sequence terms using convergence and finitely many separating <clopen sets>. Finite disjointification gives disjoint $B_n\in U_n$. The union of the even-indexed $B_n$ then belongs exactly to the even-indexed <ultrafilters>, contradicting convergence of membership in a <clopen set>.
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