Consider the function defined by
Determine whether converges or diverges, and if it converges find its limit.
This sequence diverges.
Proof. We saw in this exercise (Section 10.4, Exercise #20) that the sequence defined by diverges. We could use this to show that this sequence diverges (since for we have and so cannot approach a finite limit). For practice, we can also prove this directly from the definition by contradiction as follows. Suppose there exists a real number and a positive integer such that for all and all we have
Since is positive we know and . So, letting , we have
Adding these two expressions and using the triangle inequality we have,
This is a contradiction. Hence, the sequence does not tend to a limit .