Prove that the function $f_{n}(x) = \frac{1 - |x|^{n}}{1 + |x|^{n}}$ converges pointwise for $x\in \mathbb{R}$.

Question

I want to show that the function

$$f(x) = \frac{1 - |x|^{n}}{1 + |x|^{n}} $$

converges pointwise for all $x\in \mathbb{R}$. Furthermore, there are some intervals $(a, b)$ on which the function converges uniformly. I want to determine these intervals as well.

This is a past exam question for a final exam that I am studying for. I do not have any solutions with me, and I am new to sequences of functions. I would appreciate it if someone could help me with this question please.

Answer 1

For an outline: Do 3 cases. If $|x|<1$, it should be easy. If $|x|>1$ then divide top and bottom by $|x|$. If $|x|=1$ it should be easy again.

For uniform convergence, well, the uniform limit of continuous sequences has to be continuous. So look at the limiting function and see where it's discontinuous. That should give you an idea of, at least, where there is NOT uniform convergence.

Answer 2

Hint: First since the function is even you need to prove this statement for $x\ge 0$. This statement is clear for $x=1$. Show for $x>1$ that $$f_n(x)\to -1$$and for $x<1$ that $$f_n(x)\to 1$$