Give an example of a sequence of functions $\{f_n\}$ which converges to the constant zero function in $L^1$, but so that $f_n(x)$ does not converge to zero at any point of $[0, 1]$.
This is from a problem set on $L^p$ spaces in my undergraduate-level real analysis course. I began by noting that if $\{f_n\}$ converges to the constant zero function on $L^1$, then $\lim_{n\to\infty}\int|f_n|=0$. So I let $f_n$ be the following: $$f_n(x)=\mathbf{1}_{[0,1]}-\mathbf{1}_{(n,n+1]}$$ where $\mathbf{1}_{[a,b]}$ is the characteristic function on $[a,b]$ (and similarly for intervals that are open or half-open on either side). This allows us to conclude that: $$f_n(x)=\begin{cases} 1 & x \in [0,1] \\ -1 & x \in (n,n+1] \\ 0 & \text{otherwise}\end{cases}$$
But there is a problem with this. I established this function because I want to say that $\int|f_n|=0$ but $\lim_{n\to\infty} f_n(x) \neq 0$ for $x\in[0,1]$, and while that second part is true, the first part is not. How can I change $f_n$ so that the example actually works? Or do I have to change it entirely? If so, any advice on what $f_n$ should be?
