Proving $f_n(z)=\frac{nz}{1+n^3z^2}$ converges uniformly

Show that the sequence of functions $$f_n(z)=\frac{nz}{1+n^3z^2}$$ converges uniformly on the set $$E=[1,\infty]$$.

$$\lim_{n\to\infty}\frac{nz}{1+n^3z^2}=0$$ so it converges pointwise to 0. So I am going to check if it converges uniformly to $$0$$.

$$|\frac{nz}{1+n^3z^2}-0|\leqslant |\frac{nz}{n^3z^2}|=|\frac{1}{n^2z}|\leqslant\frac{1}{n^2}\to 0$$ as $$n\to\infty$$

since $$|\frac{nz}{1+n^3z^2}-0|$$ is majored by $$\frac{1}{n^2}$$ that does not depend on $$z$$. I conclude the function converges uniformly to $$0$$.

Questions:

Is this proof right? If not why? Which are the alternatives?