Let M be some positive integer. Does the following converge?$$\lim_{d\to+\infty}\sum_{k=M}^{d}f(\frac{k}{d})\frac{1}{d} $$ $f$ here is well defined, for example bounded by positive C and measurable on $[0,1]$.
I think it converges since $\sum_{k=0}^{M-1}f(\frac{k}{d})\frac{1}{d}\leq \frac{CM}{d}\to 0$ so: \begin{align*}\lim_{d\to+\infty}\sum_{k=M}^{d}f(\frac{k}{d})\frac{1}{d} &=\lim_{d\to+\infty}\sum_{k=M}^{d}f(\frac{k}{d})\frac{1}{d}+\lim_{d\to+\infty}\sum_{k=0}^{M-1}f(\frac{k}{d})\frac{1}{d}\\&=\lim_{d\to+\infty}\sum_{k=0}^{d}f(\frac{k}{d})\frac{1}{d}\\ &=\int_{0}^{1}f(x)dx<\infty \end{align*} Would appreciate if someone can point out any mistake.