# Convergence of sequence of functions, $f_n(x) = n^2 x(1-nx) \dots$

Doing an exercise for exam preparation, I stumbled across the following function:

$f_n(x)= n^2x(1-nx), \quad \text{if }0 \leq x \leq \frac{1}{n}$

$f_n(x)= 0, \quad \text{if } \frac{1}{n} < x \leq 1$

The task is to find the limit of this function series and to determine whether this function converges uniformly in $[0,1]$

On the one hand $\frac{1}{n}$ approaches $0$ for $n \to \infty$. So one would just have to insert $0$ in $n^2x(1-nx)$. Thereby gaining $f_n(x) = 0$ for $n \to \infty$.

On the other hand the function has a maximum for $x=\frac{n}{2}$. Putting this into $n^2x(1-nx)$ and calculating $f_n(x)$ for $n \to \infty$ afterwards one gets $f_n(x) = \infty$.

So whats correct? How does one approach such a problem?