Stylistically, it is better to call them b1 and b2, or b and b'.

You have not defined U_{a}. I assume it is the set of all upper bounds of A. Then why is the subscript lowercase a?

I assume you are considering two cases: b < b1 and b1 < b (b = b1 is impossible by assumption). Then you should say so. The problem is that theayou are using here is undefined: I have no idea what it is.

Same remark abouta.