In a Geometric Measure Theory textbook the following was written:
I cannot see how any of these equalities hold and dont believe they are obvious. If they are relatively obvious could someone please point to what information I am missing.
Here are my definitions of Minkowski dimensions (upper and lower)
Use the fact that if the $limsup$ is less than infinity for any specific value of $s,$ then the $limsup$ is zero for all strictly smaller values of $s;$ and if the $limsup$ is greater than zero for any specific value of $s,$ then the $limsup$ is infinite for all strictly greater values of $s.$ This is illustrated by the following suggestive computations, which you should be able to use as a tinplate for a formal proof.
$$\frac{N}{s^{0.2}} < \infty \;\;\;\; \implies \;\;\;\; \frac{N}{s^{0.1}} = \left(\frac{N}{s^{0.2}}\right)\left(\frac{s^{0.2}}{s^{0.1}}\right) = \left(< \infty \right)(\rightarrow 0) \; \rightarrow \; 0$$
$$\frac{N}{s^{0.1}} > 0 \;\;\;\; \implies \;\;\;\; \frac{N}{s^{0.2}} = \left(\frac{N}{s^{0.1}}\right)\left(\frac{s^{0.1}}{s^{0.2}}\right) = \left( > 0 \right)(\rightarrow \infty) \; \rightarrow \; \infty$$
["$< \infty$" means "bounded below infinity"; "$>0$" means "bounded above zero".]