Does the following converge uniformly?

Question

Does $$f_n(x)=\frac{nx}{1+n(1+x^2)}$$ converge uniformly on $\mathbb{R}$? To what? Justify.

Answer 1

Note that $${f_n}(x) = \frac{x}{{\frac{1}{n} + 1 + {x^2}}}$$

Can you work something out now? Note that the pointwise limit is $$f(x)=\frac{x}{1+x^2}$$ and that $$\begin{align} \left| {f(x) - {f_n}(x)} \right| &= \left| {\frac{x}{{\left( {1 + {x^2}} \right)\left( {1 + n\left( {1 + {x^2}} \right)} \right)}}} \right| \\ &\leqslant \frac{1}{2}\frac{1}{{1 + n\left( {1 + {x^2}} \right)}}\\ &\leqslant \frac{1}{2}\frac{1}{{n\left( {1 + {x^2}} \right)}} \\ &\leqslant \frac{1}{{2n}} \end{align} $$

because $\dfrac{x}{1+x^2}$ is at most $1/2$ and $\dfrac{1}{1+x^2}$ is at most $1$ over the real line.