Typewriter sequence ($f_{2^k+j}=1_{[j{2}^{-k},(j+1)2^{-k})}$)

On $(0,1)$, for $k\ge 0$ and $0\le j<2^k$, the displayed indicator functions form a sequence that converges to zero in every finite Lp norm while failing to converge pointwise anywhere. Indeed, for $1\le p<\infty$,Every point belongs to exactly one interval in each dyadic block, and for $k\ge 1$ 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.