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

(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)?

Shouldn't that be [1,∞)?
The first part I'm fine with and get the limit function being

as I think

Correct, except that the case when x=0 should be dealt with separately, because 1/|x| is not defined then.
For the second part I said no as I thought Sup fn was just a half. In my answers it says Sup

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

Your answer is correct, but it's probably best to include the fact that the sup is attained when x=1/n.
For iii) the answers have sup

and again while I felt it would tend to 0 I don't understand the reasoning!

[Assuming that the 1 should be ∞, as noted above.] The function is decreasing on the interval [1,∞) (because its derivative is negative), so its max value occurs at the start of the interval, when x=1, and