Behaviour of ($\overline{a}_n)_{n=1}^{\infty}$ and ($\underline{a}_n)_{n=1}^{\infty}$ I have been trying to understand the following definition and just needed some clarification.
For each bounded sequence $(a_n)_{n=1}^{\infty}$ we define the sequences ($\overline{a}_n)_{n=1}^{\infty}$   and ($\underline{a}_n)_{n=1}^{\infty}$  in the following way:
\begin{eqnarray*}
\overline{a}_n&=& \sup\left\{a_n,a_{n+1},\dots \right\},\\
\underline{a}_n&=&\inf\left\{a_n,a_{n+1},\dots\right\}.
\end{eqnarray*}
How does this definition imply that $\overline{a}_n$ is decreasing and $\underline{a}_n$ is increasing?

Also check out this image from Wikipedia's Limit Superior and Limit Inferior page. I think it may impart some intuition. Note how the top red line (sup) decreases while the bottom (inf) increases.
