TITLE: What is the difference between max and sup?
QUESTION [28 upvotes]: I am studying KS (Kolmogorov-Sinai) entropy of order q and it can be defined as
$$
h_q = \sup_P \left(\lim_{m\to\infty}\left(\frac 1 m H_q(m,ε)\right)\right)
$$
Why is it defined as supremum over all possible partitions P and not maximum? 
When do people use supremum and when maximum?

REPLY [16 votes]: A maximum is the largest number WITHIN a set. A sup is a number that BOUNDS a set. A sup may or may not be part of the set itself (0 is not part of the set of negative numbers, but it is a sup because it is the least upper bound). If the sup IS part of the set, it is also the max.