Exercise 5.2: Packing and Covering Sandwich Inequality
\begin{align} M(2 \delta) \overset{\text{(a)}}{\leq} N(\delta) \overset{\text{(b)}}{\leq} M(\delta) \end{align}
(a)
Since \(d(\theta, \theta') \leq d(\theta, \theta_i) + d(\theta', \theta_i) \leq 2 \delta\) if \(\theta, \theta'\) lie within a \(\delta\)-ball around \(\theta_i\), at most one member of any particular \((2\delta)\)-packing can lie within each of the \(\delta\)-balls around the \(N(\delta)\) points forming the \(\delta\)-covering.
(b)
If there is a point \(\theta\) with \(d(\theta, \theta_i) > \delta\) for each \(\theta_i\) constituting a \(\delta\)-packing, than \(\lbrace \theta \rbrace \cup \lbrace \theta_1, \ldots, \theta_n \rbrace\) is still a \(\delta\)-packing. Hence any maximal \(\delta\)-packing must also be a \(\delta\)-covering.