How Are Monotone Sequence Conditions and Least Upper Bound Property Equivalent?

[Scousergirl]

Prove that the monotone sequence condition is equivalent to the least upper bound theory.

I can't seem to get around how to prove that the two are equivalent. (it seems trivial).

 

[morphism]

Suppose you start from the monotone sequence condition. Given a bounded set, can you somehow use its upper bounds in some sort of useful sequence? Think of how the monotone sequence condition could apply here.

On the other hand, suppose we have the least upper bound property. Given a monotone sequence (say, a non-decreasing one) that's bounded above, is there a natural way we can transform this sequence into a bounded set? What relation could such a set's upper bound have to our sequence?