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.