Consider the function defined by
Determine whether the sequence converges or diverges, and if it converges find the limit.
The sequence diverges.
Proof. Suppose otherwise. Then there exists a real number and a positive integer such that
Since is positive, we know and . Letting we then have
So, adding these together and using the triangle inequality we then have,
But, both and are greater than for all positive integers , contradicting that for some positive integer . Hence, this sequence can have no such limit and must diverge