Exercises
[0FT]Suppose \(f\) is monotonic, show that \(\lim _{j∈ J} f(j)\) exists (possibly infinite) and coincides with \(\sup _ J f\) (if it is increasing) or with \(\inf _ J f\) (if it is decreasing).
Infer that
\begin{eqnarray*} \limsup _{j∈ J}f(j){\stackrel{.}{=}}\lim _{j∈ J} \sup _{k≥ j} f(k)\\ \liminf _{j∈ J}f(j){\stackrel{.}{=}}\lim _{j∈ J} \inf _{k≥ j} f(k) \end{eqnarray*}are always well defined.