Let $\{x_n\}$ be a bounded sequence.
a) Prove that there exists an $s$ such that for any $r > s$ there exists an $M ∈ \Bbb{N}$ such that for all $n ≥ M$ we have
$x_n < r$.
b) If s is a number as in a), then prove $ \limsup_n x_n ≤ s$.
c) Show that if $S$ is the set of all s as in a), then $\lim \sup\{x_n\} = \inf S$.
For part a) I let s = $\sup \{ X_n\}$. Let M=1, then you can prove that $X_n$ is less than or equal to $\sup \{X_n \}$.
For part b) Again let $s = \sup\{X_n\}$ then I proved that $\limsup {X_n}$ less than or equal to $\sup \{Xn \}$.
Part c is really difficult and I am not sure where to start for that problem.