This is a tough one, and I have little idea on how to prove it.
Prove that the sequence defined by is divergent.
This must be proven by contradiction.
Suppose , for some .
Let us choose for some natural number , we have that .
But this means that all but a finite number of the sequence's elements are between the left and the right ends, and the difference between these two ends is , and this is a contradiction since for any two consecutive natural numbers , so it's impossible that ALL the elements of the seq. (but perhaps a finite number of them) are within two numbers whose difference is ...