(Terras). For each k there is a finite bound given by such that implies that whenever . Consequently:
- (i) If for all , then there are no non-trivial cycles of length .
- (ii) If for all , then implies .
The existence of the bound follows immediately from (2.14), and (ii) follows immediately from this fact.To prove (i), suppose a nontrivial cycle of length exists. We observed earlier that if is the smallest element in a purely periodic nontrivial cycle of length , then and . The first part of the theorem then implies that . This contradicts the hypothesis of (i).