Uniform convergence of $f_n(x)=3x^2+\dfrac{3x}{n}+\dfrac{1}{n^2}$ for $x\in(0,\infty)$.

Question

Clearly that function converges on time to $ 3x ^ 2 $, when $ n \to \infty $ at $ (0, \infty) $. Now, for it to converge uniformly, it would be necessary to see if, $$ \sup_ {x \in (0, \infty)} | f_n-f | \to 0, \ n \to \infty. $$ Note that $$ \sup_ {x \in (0, \infty)} | f_n-f | =\sup_{x\in (0,\infty)} \left| \dfrac{3x}{n}+\dfrac{1}{n^2} \right| \ge \left | \dfrac{3n}{n} \right | = 3 $$ and this does not tend to $0$, when $ n \to \infty $. Therefore, the function does not converge uniformly. Is my proof correct?