Study the convergence of a succession of functions Study the punctual and uniform convergence of $f_n(x)$ on $A$ $$f_n(x)=\frac{x}{1+n^2 x^2} \ \ \ A=[-1,1]$$ My reasoning: Punctual convergence $\forall x \in A $ $$ \lim_{n \to +\infty} f_n(x)=f(x) \\\ f\equiv0$$ Uniform convergence It needs this propriety: $ f_{n+1}(x) \le f_n(x) $ (Dini's theorem hypothesis) So, the succession converges uniformly on $[0,1]$ Is this reasoning correct?

For an alternate proof, note that $f_n'(x)=0$ implies $x=\pm\frac1n$, and \begin{align} f_n\left(\frac1n\right) &= \frac1{2n}\\\ f_n\left(-\frac1n\right) &= -\frac1{2n}\\\ f(-1) &= -\frac1{1+n^2}\\\ f(1) &= \frac1{1+n^2}. \end{align} It follows that $$\lim_{n\to\infty}\sup_{x\in[-1,1]}|f_n(x)|=0, $$ so that $f_n$ converges uniformly to $0$ on $[-1,1]$.