The result won’t work if However I gather from your post that you want and to be both positive integers.
Start by showing that the set is not empty. This will be true in Cases 1 and 2 with For Case 3, suppose to the contrary that for all natural numbers Then, in particular, which is not possible if is a positive integer. Hence there is a natural number such that Let be the smallest such natural number. The proof is completed by taking and