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