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$ .


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

Thanks in advance!