= Typewriter sequence
{title2=$f_{2^k+j}=\mathbf1_{[j2^{-k},(j+1)2^{-k})}$}
= Dyadic typewriter sequence
{synonym}
On $(0,1)$, for $k\geq0$ and $0\leq j<2^k$, the displayed <indicator functions> form a sequence that converges to zero in every finite <Lp norm> while failing to converge pointwise anywhere. Indeed, for $1\leq p<\infty$,
$$
\|f_{2^k+j}\|_p^p=2^{-k}.
$$
Every point belongs to exactly one interval in each dyadic block, and for $k\geq1$ it misses at least one other interval in that block. Thus every point sees infinitely many ones and infinitely many zeros. Half-open intervals specify the behaviour at dyadic endpoints. This separates <norm convergence> in an <Lp space> from <pointwise convergence> and convergence <almost everywhere>.
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