Pointwise and uniform convergence.

Hi I've got some questions with answers I'm working through but can't understand some of the points and was hoping someone could explain them to me!

The question was

Let $\displaystyle fn(x) = nx/(1+n^2x^2)$

(i) Prove that (fn) converges pointwise on R and find the limit function.

(ii) Is the convergence uniform on [0, 1]?

(iii) Is the convergence uniform on [1,1)?

The first part I'm fine with and get the limit function being $\displaystyle fn \to 0$ as I think $\displaystyle fn \leqslant 1/n|x|$

For the second part I said no as I thought Sup fn was just a half. In my answers it says Sup $\displaystyle |fn(x)| \geqslant fn(1/n) \to1/2$

which is kind of what I got but don't see where they get the first part from.

For iii) the answers have sup$\displaystyle |fn(x)| \leqslant1/n \to 0$ and again while I felt it would tend to 0 I don't understand the reasoning!

Sorry if this seems stupid but I just can't see the jump in the answers to where they got what they did! If anyone could explain I'd be very grateful!