The moving indicators $\chi_{[n,n+1]}$ converge pointwise but not uniformly or in $L^1$

For \(n\geq 1\) let \(f_n\coloneqq\chi_{[n, n+1]}\colon\RR\to\RR\).

• For each \(x\in\RR\), \(f_n(x) = 0\) for all \(n > x\), so \(f_n\to 0\) pointwise on \(\RR\).

• \(\sup_{x\in\RR}\abs{f_n(x)} = 1\) for every \(n\), so \(f_n\not\to 0\) uniformly on \(\RR\).

• \(\int_\RR\abs{f_n - 0}\dx = 1\) for every \(n\), so \(f_n\not\to 0\) in \(L^1(\RR)\).