@robjohn I have been looking at the Dominated Convergence Theorem (DCT), as given
here for example. If it is applicable, then for sufficiently small $h(n)$ (i.e. large $n$), the remainder, viewed as a sequence of functions $g_{h(n)}(u)$ could be replaced by the converging function. Since it is $o(h^m)$ (presumably uniformly in $u$), the converging function seems to be $g(u)=0$.