= Unbounded typewriter sequence with vanishing L1 norm
{title2=$\|f_n\|_1\to0,\quad\limsup_nf_n(x)=\infty$}
For each $m\geq1$, 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.
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