1. ## Notation Question

How is $\Delta I$ defined for $I$ some interval on $\mathbb{R}$? Is it just $sup(I)-inf(I)$, that is, the Supremum of the interval minus the Infimum?

How is $\Delta I$ defined for $I$ some interval on $\mathbb{R}$? Is it just $sup(I)-inf(I)$, that is, the Supremum of the interval minus the Infimum?
Yes. More specifically, any interval, I, on R can be written in one of the forms (a, b), [a, b), (a, b], or [a, b]. In all of those cases $\Delta I= b- a$.