See your other thread. The proof will be the same, essentially.
I need help with the last half of the proof.
Prove that does not exist.
Assume to the contrary that this exists. Then there should exists a real number such that . Let . Then there exist . If is a real number for which , then . Choose an integer . Since , it follows that .
Let . I want to generate contradiction by showing that
With and , how can show that
Using the definition you provided:
We know that no matter how large is, there will always be an integer such that , but how do we show it?
In regard to , as we see in the graph, the range . Is it necessary that be positive? The reason I ask is that if need not be positive, then the proof is complete.