The terms of a sequence X_n take on only finitely many values a_1,...,a_k.
That is, for every n, X_n = a_i for some i (the index i depending on n). Prove X_n has a cluster point.
However, will always have a subsequence with a limit point.
Proof: Let . Because there are only finitely many points, is compact, and thus is a sequence in a compact set. So it follows from the Bolzano-Weierstrass Theorem that has a convergent subsequence (and therefore a limit point in that subsequence).