Prove that $2nx(1 + (nx)^2)^{-1}$ converges uniformly

Question

Define the function $f(x)=\frac{2x}{1+x^2}$ and let $f_n(x)=f(nx)$.

a) Show $f_n$ converges to 0 uniformly on $[1,\infty)$.
b) Show $f_n$ converges to 0 on $[0,1]$? is it uniform convergence.

I am confused here. How am I supposed to deal with uniform convergence problems for sequence of functions? What I have studied so far is about definition of convergence of sequence and definition of uniform continuity of functions and not a combination of them!

What is the strategy to answer such these questions? By the way, I am studying Goldberg textbook.

Update:
I watched some videos for different problems and I came to this part: $|f_n(x)-0|=|f_n(x)|\le \frac{1}{2}$ by finding the maximum distance.
can I say now that sequence of $f_n(x)$ converges uniformly since for all natural numbers and all points in $[0,\infty)$, has a fixed distance to $f(x)$? how should I interepret this?

Answer 1

We say that a sequence $f_n$ of functions on a set $A$ converges uniformly to a function $f$ if for every $\varepsilon > 0$ there is an $N \in \Bbb{N}$ such that $$ n > N \implies |f_n(x) - f(x)| \leq \varepsilon \quad \forall x \in A. $$

Note that $f$ is decreasing in $[1, \infty)$ and that $\frac{2n}{1 + n^2} \to 0$. Can you finish from there?