I mean a finite set of real numbers.

I can use the field axioms, order axioms, absolute value, the completeness axiom, archimedean property, and density of the rationals to prove this.

I know I need to know that such a finite set of real numbers must be bounded above. I know I need to use the completeness axiom so that the set has a least upper bound. I think that I need to show that the LUB belongs to the set??? I just cant pull it all together...