Convergence of $f_n (x)=\sqrt[3]{\rvert{x}\rvert ^3+\frac{1}{n}} \qquad x \in [-1,1]$

Question

$(f_n)$ is a succession of functions

$$ f_n : [-1,1] \rightarrow \mathbb{R} \\ \\ f_n (x)=\sqrt[3]{\rvert{x}\rvert ^3+\frac{1}{n}} \qquad x \in [-1,1] $$

Punctual convergence

$\forall x \in [-1,1] $ $$ \lim_{n \to +\infty} f_n(x)=f(x) \\ f=\rvert x \rvert $$

Uniform convergence

$ f_{n+1}(x) \le f_n(x) $

So, I can apply Dini's theorem because $[-1,1]$ is compact.

$f_n$ converges uniformly on $[-1,1]$

Is it correct?

Thanks!

Answer 1

For the punctual convergence consider the two cases $x \le 0$ and $x > 0$. You should find $f=|x|$.

But you are applying correctly Dini's theorem. This part is correct. But to justify a bit more you should say that $f$ and $f_n$ are continuous.