Show that $$f_n(x)=\frac{n^2x^2}{2+n^3x^3}$$ converges pointwise on $[0,\infty)$, but does not converge uniformly.
Here's what I have so far: I've shown that $\lim_{n\to\infty} f_n(x) = 0 = f(x)$ for all $x\in[0,\infty)$, so $f(x)=0$ is the pointwise limit.
To show that it does not converge uniformly, I'd like to invoke the result that there exists a sequence of points $\{x_n\}$ in $[0,\infty)$ such that $|f_n(x_n)-f(x_n)|\geq \lambda$ for some constant $\lambda>0$. This boils down to $$ \left|\frac{n^2x^2}{2+n^3x^3} - 0\right| = \frac{n^2x^2}{2+n^3x^3} $$ since $x\geq 0$. Now I need help finding a sequence of points such that this is greater than or equal to some constant. Any hints here would be greatly appreciated. Thanks.
The fact that you have expressions $n^2x^2$ and $n^3x^3$ suggest $x_n=1/n$.