Riemann Sum $\epsilon$ Criterion 

In iii) I do not understand why the solution lets $\Delta_\epsilon$ be $\Delta_2$ when $\epsilon\geq 8$ i.e $>2$
 A: If I understand your question, it's confusing you that part (iii) says "find a partition" and then in the solution they're using two different partitions. The key here is that part (iii) of the question says "Given an $\epsilon > 0$, find a partition..." which means that if someone hands you an $\epsilon > 0$, the partition you choose can depend on the $\epsilon$ you are given. The partitions we use for two different values of $\epsilon$ can be completely different and in many problems of this type they will be. 
The author here splits the problem into two cases: $\epsilon < 8$ and $\epsilon \geq 8$. If we are given any $\epsilon < 8$, we know how to find a partition that does what we want using $\Delta_{\epsilon}$ (which is a function of $\epsilon$). If $\epsilon \geq 8$, then we don't even need the partition to depend on $\epsilon$ because we can just use $\Delta_2$ as that happens to work in all cases. We don't need the partition to depend on $\epsilon$ for values of $\epsilon \geq 8$ because $\Delta_2$ because $S(f, \Delta_2) = 1$ and we certainly have $1 \leq 8 \leq \epsilon$ for this range of $\epsilon$ values. 
A: It is not that there are two different partitions, but that apparently there are two rules to generate a partition depending on whether $\epsilon<8$ or not. The exercise is asking you to construct a partition for each $\epsilon>0$, which means that you need to construct a lot of partitions if you imagine that $\epsilon$ is varying.
