What is the difference between max/min and sup/inf.

Question

Say there exists a series {A}, which has an upper bound, meaning it has a supremum, call is "a" ( sup(A) = a ). My question is, is {A}≤a or {A}<a? My confusion arises from the fact that we know that if "a" is included in {A} then it is also its maximum, meaning {A}≤a, does that still stand for the supremum? I know that if there is a x∈A, st x≤a than a is the maximum of {A}, so consequently if sup(A) would have the same definition, meaning {A}≤a, wouldn't it be a maximum too? I would really appreciate some help to clear up this confusion, thank you.

Answer 1

If a set $A$ has an upper bound, then $A$ has a least upper bound (in $\mathbb{R}$). Then for every $a \in A$, the supremum $\sup A$ is an upper bound of $A$ so $a \leq \sup A$. So, to abuse notation a bit, $A \leq \sup A$.

The supremum is meant to generalize the idea of a maximum. You can show that if the set contains an upper bound, then supremum and maximum coincide. In particular, on finite sets the maximum and the supremum are precisely the same.