Originally Posted by

**shmounal** 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)? Shouldn't that be [1,∞)?

The first part I'm fine with and get the limit function being $\displaystyle fn \to 0$ as I think $\displaystyle fn \leqslant 1/n|x|$ 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 $\displaystyle |fn(x)| \geqslant fn(1/n) \to1/2$

which is kind of what I got but don't see where they get the first part from. 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$\displaystyle |fn(x)| \leqslant1/n \to 0$ 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 $\displaystyle \color{blue}f_n(x)$ 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 $\displaystyle \color{blue}f_n(1) = n/(1+n^2) < 1/n.$