Exercise
47
[0BQ] We fix a real valued sequence \(a_ n\). Now consider the definition of [20F] setting \(I=β\) and \(x_ 0=β\), so that neighborhoods of \(x_ 0\) are sets containing \([n,β)=\{ mββ:mβ₯ n\} \); with these assumptions show that you have
\begin{align} \limsup _{nβ β} a_ n =& \inf _ n \sup _{mβ₯ n} a_ n= \lim _{nββ} \sup _{mβ₯ n} a_ n~ ~ , \nonumber \\ \liminf _{nβ β} a_ n =& \sup _ n \inf _{mβ₯ n} a_ n= \lim _{nββ} \inf _{mβ₯ n} a_ n~ . \label{eq:def_ limsup_ liminf_ succ}, \end{align}