I'm trying to study this sequence of functions:
$f_n(x) = nxe^{-nx}$ for $x \in [0,1]$
i ended up with the
point convergence -> $f(x) = 0$
uniform convergence The function doesn't have the uniform convergence to $f(x) = 0$ in $[0, 1]$. Becouse during the study of the convergence if found that the $sup_{x \in [0,1]}|f_n(x) - f(x)| = \frac{1}{n}$ and for this $x=\frac{1}{n}$ the $\lim\limits_{n \to +inf}e^{-1} \not= 0$
I think that that's ok but I know that a correct and complete study of the sequences of functions include the analisys of the uniform convergence in other sub-ranges $[\alpha, \beta] ⊂ [0, 1]$. I don't know how to do this in a step-by-step way, but i think that in this case the uniform convergence is present in
$[0, a] ⊂ [0, 1]$ with $a<\frac{1}{n}$
$[b, 1] ⊂ [0, 1]$ with $b>\frac{1}{n}$
where $\frac{1}{n}$ is the max value assumed by the function at index $n$ of the sequence. Am i right or thats a wrong idea?