A function $\rho$ is given in Michael Field's $Essential$ $Real$ $Analysis$ as follows:
$\rho(A,B)=\sup\limits_{a\in A}d(a,B)=\sup_\limits{a\in A}\inf\limits_{b\in B}d(a,b)$, where $d$ is the euclidean metric and $A$ and $B$ are compact subsets of $\mathbb{R^n}$.
$d(a,B)$ is defined to be $\inf\{d(a,b)\mid b\in B\}$ for the point $a\in A$.
I'm quite lost with this. $\inf d(a,b)$ is simply a number so $\sup$ of it is the number itself. So taking the $\sup$ makes no difference and $\rho$ seems to be the same as $d(A,B)=\inf\{d(a,b)\mid a\in A,\,b\in B\}$.
What is it that I don't understand? Why does he take the $\sup$ and what difference does it make? He writes that "roughly speaking", $\rho$ gives the greatest distance of the points $a\in A$ and $b\in B$. I can catch the $\sup d(a,B)$ as the maximum distance: going through all the $a\in A$ and taking the $\sup$. But the $\sup\inf$ -part of the equation looks like the minimum distance given by $d(A,B)=\inf\{d(a,b)\mid a\in A,\,b\in B\}$: going through all the pairs $a\in A$ and $b\in B$ and taking the $\inf$. That gives a number and taking the $\sup$ of it doesn't change anything.
Yes, $d(a,b)$ is simply a number, for each $a$ and each $b$. But, if you fix $a$, then $\{d(a,b)\mid b\in B\}$ is not a number; it is a set of numbers. And the author defined $d(a,B)$ as the infimum of that set.