Talking about $f_n(x)=\frac{nx}{1+nx^2}$

Question

Let $$f_n(x)=\frac{nx}{1+nx^2}$$

for $x\in [0,\infty)$

Find $f(x)=\lim f_{n}(x)$

Does $f_n\to f$ uniformly on $[0,1]$

Does $f_n\to f$ uniformly on $[1,\infty)$

I have solved the first and third question. For third question, answer is yes.

For first question

$f(x)=\frac{1}{x}$ and limit does not exist at $0$.

Now how can I approach to solve $2$ question?

I tried using the following idea $M_n=\sup_{{x\in (0,1]}}|f_n-f|=|\frac{1}{1+nx^2}|$

Now the problem is $\sup$ is attained at $0$ but the function is not defined there. I am stuck here.

Second thing, when I googled this question I found this(http://homepages.warwick.ac.uk website):

Looks like the second bullet point is the solution of my question but what is the idea. What technique are they using?

Answer 1

Hint: the uniform limit of a sequence of continuous functions is a continuous function.