You are extremely close to the solution. The last brick in the proof comes from the more explicit version of Bezout's lemma. Given for , not only do there exist an infinity of integer solutions , but they can also be enumerated by
In your equation, find a solution to and call it . You will now notice that for any integer , another solution exists . So for a high enough , and will both be positive, and by increasing arbitrarily, all higher solutions will increase positively without limit.