Prove that a function can be represented as Infinite Darboux sums $f$ is positive and decreasing monotonously.
Both $\int _0^\infty f(x)dx$  and $\sum_{n=1}^\infty f(nk)$ converges. 
Prove that:
$$\lim_{k^+\to 0} k\sum_{n=1}^\infty f(nk)= \int _0^\infty f(x)dx$$
What I tried to do:
I feel as if I can "see" the proof but I'm unable to write it. 
It's like I'm being asked to prove that Upper Darboux Sum converges to the integral when the intervals become narrow (lets assume $k$ is the width of Darboux's rectangles)
Or maybe something regards Riemann sums?
Only that all of these sentences where taught for finite intervals. Maybe show somehow the tail is approximately zero?
Someone, Help?
:-)
 A: If you have no problem with infinite "Darboux sums", you can simply observe that, since $f$ is monotonically decreasing,
$$k\sum_{n=1}^\infty f(nk)\tag{1}$$
is a lower Darboux sum for the integral $\int_0^\infty f(x)\,dx$, and $(1)$ is also an upper Darboux sum for the integral $\int_k^\infty f(x)\,dx$. That yields
$$\int_0^\infty f(x)\,dx - k\cdot f(0) \leqslant \int_k^\infty f(x)\,dx \leqslant k\sum_{n=1}^\infty f(nk) \leqslant \int_0^\infty f(x)\,dx.\tag{2}$$
Letting $k \searrow 0$ in $(2)$ shows $\lim\limits_{k\searrow 0} k\sum\limits_{n=1}^\infty f(nk) = \int_0^\infty f(x)\,dx$.
If you don't want to argue with infinite Darboux sums, you can still argue that for every $N\in \mathbb{N}$, the finite sum
$$S_N = k\sum_{n=1}^N f(nk)$$
is a lower Darboux sum for the integral $\int_0^{Nk} f(x)\,dx$ and an upper Darboux sum for the integral $\int_{k}^{(N+1)k} f(x)\,dx$, so
$$\int_0^{(N+1)k} f(x)\,dx - kf(0) \leqslant S_N \leqslant \int_0^{Nk} f(x)\,dx \leqslant \int_0^\infty f(x)\,dx.\tag{3}$$
Letting $N\to\infty$ for $S_N$ in $(3)$ shows
$$\int_0^{(N+1)k}f(x)\,dx - kf(0) \leqslant k \sum_{n=1}^\infty f(nk) \leqslant \int_0^\infty f(x)\,dx\tag{4}$$
for all $N\in\mathbb{N}$. Now letting $N \to \infty$ on the left hand side of $(4)$, we again obtain $(2)$. And once more, squeezing yields
$$\lim_{k\searrow 0} k\sum_{n=1}^\infty f(nk) = \int_0^\infty f(x)\,dx.$$
