Supremum vs. Infimum

Question

I learn best with visuals, so I am struggling to understand why the Supremum is called the "Least Upper Bound" and not the "Greatest Upper Bound", like the Infimum is called the "Greatest Lower Bound". Why is the Supremum "Least" and not "Greatest"?

Answer 1

Take $A\subseteq \mathbb{R}$ and suppose that it is bouonded from above. This means that there exists $M\in \mathbb{R}$ such that $a\leq M$ for all $a\in A$. We call such an $M$ an upper buond for $A$. It can be proved that, by completeness of real numbers, the set of the upper bounds of $A$ has a minimum, and this minimum is called the supremum of $A$. An analogous construction is done for the infimum of sets that are bounded from below.

Thinking to this construction should give you the answer.