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 U_{a. }b<= a for all element a that are in set U_{a }

b1 is a element in U_{a}

b1<=a for all element a that are in U_{a }i) b<b1<a

Only b can be sup A. Contradiction

ii) b1<b<a

only b1 can be sup A. Contradiction.

Therefore b=b1.