Prove that the sequence converges to 0.

Here's the solution:

http://img517.imageshack.us/img517/7235/81768116.gif

Here's my problem: I don't understand why they wrote !

We know that

Therefore

And so we should take:

Isn't that right???

Also, I don't understand why they first omitted from the denominator when they could omit instead. Because I think goes to zero a lot quicker. I appreciate some explanation.