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?

If $A\subset B$ then $\sup A\leq \sup B$ and $\inf A\geq \inf B$. For example if $A=\{2,1,1/2,1/4,...\}$ and $B=\{4,2,1,1/2,1/4,...\}$ then $\sup A=2<4=\sup B$.......If $A\subset B$ then any upper bound for $B$ is also an upper bound for $A$.So the least of the upper bounds for $B$ is one of the upper bounds for $A$ but maybe not the least of them.Similarly for lower bounds.
