Do you know that the positive integers are well ordered?
Let .
Let be the first term in .
Oops! How did I do that mistake
Then, let me correct.
Suppose that for every , . However, since is fixed, we can find such that , i.e.,
Then, we know that , similarly , and by following similar arguments, we show that .
This shows that must be negative, and thus is a contradiction.
But I have to say that this is not a nice proof.
Thanks to Plato for pointing out my mistake.