Let
be a bounded sequence, and for each
let
$\displaystyle sup${
>} and $\displaystyle t_n$ := $\displaystyle inf${
>}. Prove that $\displaystyle (s_n)$ and $\displaystyle (t_n)$ are monotone and convergent. Also prove that if $\displaystyle lim(s_n) = lim(t_n)$, then $\displaystyle (x_n)$ is convergent.