**Problem:**

prove that the sequence of functions defined by:

converges uniformly on where .

Does converges uniformly on

Here is what I did: I proved that the pointwise limit of is the function . Then, in order to prove the uniform convergence, I need to prove that as and that's where I am stuck. In the book, there is a hint saying that for x in the mentioned interval , and using the inequality ,

we get:

I don't understand how the book got this inequality based on . Can anyone give me a detailed proof how to get that inequality given by the book? From that point, I can easily prove the uniform convergence.

For the uniform convergence on R, there is hint saying that I should use to get:

Any help please how can we derive the last inequality too? because from this inequality, I can easily prove the non uniform convergence on R.