Prove that $f_n(x): x\in\mathbb{R}\mapsto \frac{x}{(1+x^2)^{n}}$ converges pointwise to null function Show that sequence of function :
$$f_n(x): x\in\mathbb{R}\longmapsto \dfrac{x}{\left(1+x^2\right)^{n}}$$
converges pointwise to null function 
since we found from the following link that :

Show $\frac{x}{\left(1+x^2\right)^{n}}\underset{{n\to +\infty}}{\sim} \frac{1}{x^{2n-1}}$ for every $x>1$

is false 

https://books.google.com/books?id=Ihh2uOXnRQcC&hl=fr&pg=PA97&#v=onepage&q&f=false


*

*since the proof provided by the book Is there alternative proof


Of course $f_n(0)=0$ for any $n$ and there is nothing to prove. Suppose $x\neq 0$. Then $1+x^2=:q>1$ and 
$$
\lim_{n\to +\infty} \frac{1}{q^n}=0,
$$
therefore
$$
\lim_{n\to +\infty}f_n(x)=x\cdot \lim_{n\to +\infty} \frac{1}{q^n}=x\cdot 0=0,
$$
