Let $H$ be the non-empty^{1} 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. ↩