Demonstrating pointwise convergence

Question

I can't seem to find a way to demonstrate that the sequence of functions $$f_n(x)= \left\{ \begin{array}{c} 4n^2x\qquad if \qquad 0\le x\le \frac1{2n} \\ -4n^2x+4n\qquad if \qquad \frac1{2n}\lt x\lt \frac1{n} \\ 0\qquad if \qquad \frac1{n}\le x\le 1 \\ \end{array} \right. $$ converges pointwise. I know that I need to find $f(x)$ but I don't know how exactly. Any help would be appreciated.

Answer 1

First note that $f_n(0)=0$ for all $n$. And if $0<x\leq 1$ then there is some positive integer $N$ such that $\frac{1}{N}\leq x$, so $f_n(x)=0$ for $n\geq 0$. Therefore the pointwise limit is $0$.

In general, it can be helpful to sketch the graphs of the first few $f_n$. In this case the portion of the graph where $f_n$ is non-zero is a triangle which becomes increasingly tall and thin as $n$ increases.