Point-wise convergence of function

Question

I have been trying out some questions on sequence of functions.In one of those questions,I am supposed to find the point-wise limit of the following sequence of functions defined on [$0,1$] as $$f_n(x)=\begin{cases} n^2x, \text{if 0$\le$$x$$\le$$\frac{1}{n}$}\\ -n^2x+2n, \text{if $\frac{1}{n}$$\le$$x$$\le$$\frac{2}{n}$}\\ 0, \text{if $\frac{2}{n}$$\le$$x$$\le$1} \end{cases}$$ Given this,I am unable to find the limit of f$_n$(x) as n tends to infinity.

Help please!

Answer 1

The answer is that it converges to zero everywhere.

Proof:
For $x = 0$, $f_n(x) = 0$ for all $n$. For any other $x$, take $n \ge \frac2x$, for which $f_n(x) = 0$ because $\frac2n \le x \le 1$.