I can point out that your statement, "since the hypothetical ancestral possibilities are infinite, that no number can be excluded from a possible result" is incorrect. There are an infinite number of even numbers (so the "hypothetical ancestral possiblities" are infinite) but you certainly can't say "no number can be excluded from a possible result: any odd number can be excluded.
Saying that a set contains an infinite collection of numbers does NOT mean that "every number is in it" nor that "we cannot exclulde any number". Another, more relevant example: the set of all numbers of the form m/2^n for m and n positive integers contains an infinitely many numbers but does not contain the number 1/3 because 3 is not 2^n for any n.
It would be a rather complicated mathematical proof but the basic idea is just what your brother said: If A is purely Italian and B is purely some other ethnicity, then any child will get half his genes from A and half from B and so will be "A/2+ B/2= (A+B)/2" "Italian". If A and B are each "part Italian", you still have (A+ B)/2 and so a "2" in the denominator. No matter how far you go back, 100 generations or 1000, in each step, you will be dividing by a new power of 2. Because you start with integer number of genes ("pure Italian" or "pure non-Italian"), the only thing that ever happens to the denominator is "dividing by 2". You can only have a "power of 2" in the denominator, never a "3".