Prove uniform convergence of $f_n = \frac {x} {1+x^n}$

Question

$$f_n = \frac {x} {1+x^n} , x \in [0,\infty)$$

I'm not sure what cases I have here. I thought maybe to check whenever n is odd or not ...

thanks for the help !

Answer 1

The pointwise limit is $$f(x) = \begin{cases} x&0\leq x<1\\ 0.5 &x=1\\ 0 &x>1\end{cases}$$

Can you conclude now?

$\textbf{Added}$:

Here, each $f_n$ is continuous but $f$ is not continuous (at $1$ ), so the convergence is not uniform. More generally, the convergence cannot be uniform on any domain containing $1$.