For $n ∈ N$, define the formula,
$$f_n(x)= \dfrac{x}{nx^{2}+1}$$
Prove that $f_n$ converges uniformly on $\mathbb R$, as $n \to\infty$.
I know that the definition says $f_n$ converges uniformly to f if given $∀ ϵ>0$, $∀ n≥N$, such that $\lvert f_n(x)−f(x)\rvert \lt \epsilon$, $\forall n \ge N$ and $\forall x \in \mathbb R$.
But I can´t find an $n≥N$ such that $|f_n(x)−f(x)|<ϵ $
Any help is appreciated!
HINT $1$: Compute the maximum of the generic term $f_n$ and see where it converges and, if such convergence is independent of $x$
HINT $2$: observe the puntual limit is $0$ and bound $f_n$ from above with a constant dependent of $n$ and independent of $x$