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? 
 A: Yes, your reasoning is correct: a monotone sequence converging pointwise to a continuos function converges uniformly. In case you are interested in a proof, you can find one here. The same reasoning can be applied to prove uniform convergence in $[-1,0]$, where the sequence of functions is monotonically increasing to $0$.
Just a side note: it is not true that "it needs" that property, as you say. Indeed, you can prove prove uniform convergence directly from the definition. This can be done considering the derivative of $f_n$ and looking for where it attains the maximum distance from the limiting function. But derivatives never bothered you anyway.. :)
A: 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 $[-1,1]$.
