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

which implies

Therefore,

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