Given a bounded sequence $\{a_n\}_{n=0}^{\infty}$ and the sequence $\{b_m\}_{m=0}^{\infty}$ defined by: $b_m=\sup \{a_m, a_{m+1},...\}=\sup \{a_n|n \ge m\}\;.$
Prove that there exists $L=\lim_{m\to\infty} b_m$ and that $L$ is a subsequential limit of $\{a_n\}_{n=0}^{\infty}$. ( is it the limsup of $a_n$?)
Thank you very much.