The problem: Show that if $a>0$, then the sequence $f_n(x) = (n^2x^2e^{-nx})$ converges uniformly on the interval $[a,\infty)$ but not on $[0,\infty)$
My Solution: On the interval of $[0,\infty)$ we notice that $f_n(x)$ attains a maximum at $2/n$ by taking the derivative and setting it equal to zero. Since $f_n(2/n) = 4e^{-2}$, we know that our function cannot converge uniformly on $[0,\infty)$ since $||f_n||\not\rightarrow 0$. On the interval of $[a,\infty)$ $f_n$ does converge uniformly, because for large $n$, $2/n \rightarrow 0$, and therefore $a>2/n$ and thus $||f_n|| = n^2a^2e^{-na}$ which tends towards zero, and thus converges uniformly.
My question : It seems that the problem is at $x=0$ because removing this point allows for uniform convergence. What I dont get is why $x=0$ causes a problem to begin with. If we analyze $f_n$ pointwise at $x=0$, dont we have that $f_n(0) = 0$ for all $n\in \mathbb{N}$? Im conceptually lost as to where the problem arises by retaining the $x=0$ value.
Thanks for your help!