EDB — 0FT

Exercises

1. [0FT]Suppose $f$ is monotonic, show that $\underset{j\in J}{lim}f(j)$ exists (possibly infinite) and coincides with $\underset{J}{sup}f$ (if it is increasing) or with $\underset{J}{inf}f$ (if it is decreasing).

   Infer that

   $$\begin{array}{r}\underset{j\in J}{lim sup}f(j)\stackrel{.}{=}\underset{j\in J}{lim}\underset{k\ge j}{sup}f(k)\\ \underset{j\in J}{lim inf}f(j)\stackrel{.}{=}\underset{j\in J}{lim}\underset{k\ge j}{inf}f(k)\end{array}$$

   are always well defined.