Let H be the non-empty1 set of all accumulation points of a bounded real sequence (a_n)_{n\in\mathbb N}. The limit superior of a_n equals the supremum of H, formally \overline{\lim_{n\to\infty} a_n}=\sup H.
Proofs: 1
Please note that H is not empty due to the theorem of Bolzano-Weierstrass, since the sequence is bounded by hypothesis. ↩