EDIT: Given $f_n(x)=\frac{x}{1+x^2}-\frac{(x^2+1)x}{1+(n+1)^{2}x^{2}}$
Show that $f_n(x)$ converges pointwise to some $f(x)$ but that the convergence is not uniform.
EDIT: I tried finding $\varepsilon$ and $x$ so that $\forall$ $n_0\in\Bbb{N}$ $\exists$ $n$$\ge$$n_0$ $\exists$ $x$$\in$$\Bbb{R}$ so that $\mid$$f_n(x)-f(x)$$\mid$$\ge$$\varepsilon$. But I could not do that. Am I on the right track? Or is there a better way?
HINT:
For $x>1$, we have
$$\left|\frac{(1+x^2)x^2}{1+(n+1)^2x}\right|>\frac{(1+x^2)x}{x^2+(1+n)^2x^2}=\frac{(1+x^2)/x}{1+(1+n)^2}>\frac{x}{1+(1+n^2)} \tag 1$$
Can you find a number $\epsilon >0$ and some value of $x \in [1,\infty)$ such that $\epsilon$ is smaller than the right-hand side of $(1)$ for all $n\ge 1$?