Unbounded typewriter sequence with vanishing L1 norm ($\Vert f_n{\Vert }_1\to 0,{\mathrm{limsup}}_n{f}_n(x)=\infty$)

For each $m\ge 1$, enumerate the $m^2$ half-open intervals $[j/m^2,(j+1)/m^2)$ partitioning $[0,1)$ and use the functions $m$ times their indicator functions. Concatenate these finite blocks in increasing $m$. Every function in block $m$ has L1 norm $1/m$, which tends to zero. Every $x\in (0,1)$ is in one interval of every block, so the values at that point have a subsequence increasing to infinity. Thus small integral norm does not bound the pointwise upper limit, even when every function is nonnegative. In contrast, Fatou's lemma prevents a strictly positive lower limit everywhere on an interval when the norms tend to zero.