Let D be an ample R-divisor, is the round down [mD] very ample for any sufficiently divisible number m?


I think it's true. But I do not know how to arrange an argument.



Is [mD] very ample if D is ample?
I am not sure if this is the shortest proof, but I think that it is a proof. 


Let $A=$ very ample line bundle. After replacing D by a multiple, you may assume that 
$$C=D - K\_X - (n+1) A$$ is ample where $n=\dim X$. 


By Angehrn and Siu, we know that $$K\_X+(n+1)A + \text{(ample line bundle)}$$ is very ample. 


Now 
$$[mD] = K\_X+ (n+1) A + C + [mD] - D$$ 


and you just need to make sure that $[mD]+C-D$ is ample if $m\gg 0$. This is easy to check by Diophantine approximation.


