Determine whether a sequence of functions converges uniformly

Question

The function $f_n(x)=3n^3(x-1/n)^2$ for $x\in[0,2]$ is given, and I need to show whether the sequence of functions $(f_n)_{n\in\Bbb{N}}$ converges uniformly to $f(x)=\lim\limits_{n \to \infty}f_n(x)$, which I have determined to be $3x^2$. It's given that $(f_n)_{n\in\Bbb{N}}$ converges pointwisely, but I'm not sure whether that information is useful in this case, and I'm not sure how to proceed to determine whether it's uniformly convergent or not. Any help would be appreciated

Answer 1

Ok! So, in the comments we discovered $f_n(x) = 3n^3 (x- \frac{1}{n})^2$ only if $0 < x \leq \frac{2}{n}$ and $f_n(x)=0$ otherwise.

Now this clearly converges pointwisely to $0$. (For any $x>0$ for $n$ big enough $x > \frac{2}{n}$)

But $$ ||f_n-0|| = \sup\limits_{x \in (0,\frac{2}{n}]} |f_n(x)| \geq f_n(\frac{2}{n}) = 3n$$ and thus the convergence is not uniform.