proof of theorem on equivalent valuations
Assume that the valuations and are equivalent. Let be an element of such that . Because the valuations are assumed to be equivalent, it is also the case that . Hence, there must exist positive constants and such that and .
We will show that show that for all by contradiction.
Let be any element of such that . Assume that . Then either or . We may assume that without loss of generality.
Since , there exists an integer such that . Let be the least integer such that . Then we have
Since , this implies that
which is impossible because the two valuations are assumed to be equivalent.
|Title||proof of theorem on equivalent valuations|
|Date of creation||2013-03-22 14:55:40|
|Last modified on||2013-03-22 14:55:40|
|Last modified by||rspuzio (6075)|