= Solution
Let $(V_n)$ be a <convergent sequence> in the <Stone-Čech compactification of the natural numbers>, with limit $V$. Each fixed $W\ne V$ occurs only finitely often: the <Hausdorff> property gives a <neighbourhood> of $V$ excluding $W$, and eventually the sequence is in that <neighbourhood>.
If $(V_n)$ were not eventually equal to $V$, infinitely many terms would differ from $V$. They could not take only finitely many distinct values, because each such value occurs only finitely often. Recursively select a <subsequence> of pairwise distinct terms, all different from $V$. It still converges to $V$, contradicting the preceding parts. Hence
$$
\boxed{V_n=V\text{ for all sufficiently large }n.}
$$
This establishes <no nontrivial convergent sequences in the Stone-Čech compactification> for arbitrary sequences, including ones with repetitions.
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