proof of theorems in additively indecomposable
- •
is closed.
Let be some increasing sequence of elements of and let . Then for any , it must be that and for some . But then .
- •
is unbounded
.
Consider any , and define a sequence
by and . Let be the limit of this sequence. If then it must be that and for some , and therefore . Note that is, in fact, the next element of , since every element in the sequence is clearly additively decomposable.
- •
.
Since is not in , we have .
For any , we have is the least additively indecomposable number greater than . Let and . Then . The limit case is trivial since is closed and unbounded, so is continuous.