Let be bounded below, and define is a lower bound for . Show that .

This is what I have:

Let . Let be any lower bound for . It follows that . Since for all , is an upper bound for .

Now I need to show that for any other upper bound for , . I think this has to do with showing that any upper bound for B is an element of A, but I can't figure out how to show this.

Thanks for the help,

Ultros