Uniform convergence of $f_n(x)=nx^{n-1}$ on $[0,1)$.

Question

I wonder if there is a way to prove the sequence $\{f_n(x)\}$, where $f_n(x)=nx^{n-1},0\le x<1$, to be uniformly convergent? Thank you very much in advance.

Answer 1

It converge pointwise to $0$, but $$\sup_{x\in[0,1[}|f_n(x)|=n\underset{n\to \infty }{\longrightarrow }+\infty $$ therefore it can't be uniformly convergent.

Answer 2

Observe that $$\int_{[0,1)}f_n(x)\ \mathsf dx = \int_{[0,1)} nx^{n-1}\ \mathsf dx = 1 $$ for all $n$, but $\lim_{n\to\infty}nx^{n-1}=0$ for all $x\in[0,1)$. Since $$\lim_{n\to\infty}\int_{[0,1)}f_n(x)\ \mathsf dx \ne \int_{[0,1)}\lim_{n\to\infty} f_n(x)\ \mathsf dx, $$ we conclude that $f_n$ does not converge uniformly on $[0,1)$.