If an element $a \in A$ is an upper bound for $A$, then $a = \sup A$.

Question

Prove that if an element $a \in A$ is an upper bound for $A$, then $a = \sup A$.

To prove this, do I need two cases and follow the definitions of the infimum and supremum?

Answer 1

Hint: To show $a$ is the supremum of $A$, you need to show two things:

• $a$ is an upper bound for $A$. (Given.)

• If $b$ is any upper bound for $A$, then $b \ge a$. (For this part use the fact that $a \in A$.)