Moving-spike obstruction to interchanging a limit and an infinite sum
= Moving-spike obstruction to interchanging a limit and an infinite sum
Let $f(t,n)=\mathbf1_{[n,n+1)}(t)$ for $t\geq1$. For every fixed $n$, $f(t,n)\to0$ as $t\to\infty$, but $\sum_nf(t,n)=1$ for every $t\geq1$. Pointwise convergence without monotonicity or domination therefore does not justify interchanging a limit and an infinite sum.