Uniform convergence in closed segment

Question

Let $f_n(x)=\frac{2nx+1}{n+nx^2}$. I want to prove that it is uniform converging in $[0,3]$.
The pointwise limit function is $f(x)=\frac{2x}{1+x^2}$.
I was able to find a supremun to $|f_n(x)-f(x)|$ which is $\frac {1}{n} \rightarrow 0$.
Does this prove that $f_n(x)$ converges in $[0,3]$?

Answer 1

The fact $\sup_{x\in[0,3]}|f_n(x)-f(x)|$ converges to $0$ is indeed enough, and actually equivalent to the uniform convergence on $[0,3]$. This is an instructive exercise to show it with $\varepsilon$'s.

And indeed, in your context, the supremum is equal to $1/n$ (reached at $0$).