prove for all real numbers $x$ we have the following inequality $$\sum_{k=1}^n \left(\frac{{\lfloor}kx{\rfloor}}{k} \right)\le {{\lfloor}nx{\rfloor}} $$
i can easily prove for $n=1$,how do i approach for $n=m$ and henceforth $n=m+1$. Or am i approaching this wrong--by trying with induction??