This is a rather easy problem but I'm not very sure if my proof is correct.
Please check for any lapses in logic.
Thank you in advance.
Show that for any subset A of R, sup A is unique.
pf) Suppose two sup A exist. b and b1
Def) b is a element in Ua.
b<= a for all element a that are in set Ua
b1 is a element in Ua
b1<=a for all element a that are in Ua
Only b can be sup A. Contradiction
only b1 can be sup A. Contradiction.