proof of equivalence of definitions of valuation
We will start with a lemma:
Lemma Let be a valuation according to the definition of the parent entry with constant . Then .
Proof: We will start with the case where is a power of two: We shall prove that by induction. The assertion is certainly true when by hypothesis. Assume that it also holds when . Then we have
To deal with the case where is not a power of two, we shall pad the sum with zeros. That is to say, we shall define when . Let be the greatest integer such that . Then and we have
Corollary Let be a valuation according to the definition of the parent entry with constant . Then, if is a positive integer, .
Proof: Write . Then, by the lemma,
However, since is a valuation since and so , hence
Having established this lemma, we will now use it to prove the main theorem:
Theorem If is a valuation according to the definition of the parent entry with constant , then satisfies the identity
Proof: Let be a positive integer. Then we have
Using the lemma, we can bound this:
Using the corollary to the lemma, we can bound the binomial coefficient to obtain
Using the obvious inequality
If either or , then the theorem to be proven is trivial. If not, then and we can divide by to obtain
By the theorem on the growth of exponential function, it follows that this inequality could not hold for all unless
in other word, unless .
|Title||proof of equivalence of definitions of valuation|
|Date of creation||2013-03-22 14:56:00|
|Last modified on||2013-03-22 14:56:00|
|Last modified by||rspuzio (6075)|