There is a subtle but important difference. To be more precise, consider the following two statements: 1) There exists some n such that all integers >= n satis

There is a subtle but important difference. To be more precise, consider the following two statements: 1) There exists some n such that all integers >= n satisfy the desired property. 2) For n = [a specific constant], all integers >= n satisfy the desired property. These two statements are not the same, but both imply that there are a finite number of counterexamples. The second one is stronger, since we could prove the statement by enumerating all k < n and checking the statement for each such k; if all these checks pass, then the statement is correct. This strategy does not work for the first strategy since we do not know n, only that such an n exists. In particular, there could be a non-constructive proof that establishes existence of such an n without providing any way to compute such an n. From the discussion, it does sound like this paper is proving (2), not (1).