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?

Note that 
$$\bar a_2 = \sup\{ a_2, a_3, \cdots \} \le \sup\{ a_1, a_2, a_3, \cdots \} = \bar a_1$$
as the set $\{ a_2, a_3, \cdots \}$ is contained in $\{ a_1, a_2, a_3, \cdots \}$. Similarly we have $\bar a_{n+1} \le \bar a_n$ for all $n$. 
