Least upper bound property of an ordered field

[Matherer]

I am trying to understand the following theorem:

An ordered field has the least upper bound property iff it has the greatest lower bound property.

Before I try going through the proof, I have to understand the porblem. The problem is, I don't see why this would be true in the first place... I can have an upper bound without a lower can't I? Can someone explain?

 

[bpet]

... I can have an upper bound without a lower can't I?

(this belongs in the Algebra forum)

Hint: If S is such a set, consider -S.

 

[HallsofIvy]

Yes, you can have an upper bound without a lower bound. But that is NOT what either the "least upper bound property" or "greatest lower bound property" say!

The "least upper bound property" says "If a set has an upper bound, then it has a least upper bound". The "greatest lower bound property" says "If a set has a lower bound then it has a greatest lower bound"