I guessed what the supposed inequality is, but this guess is a false inequality. I've attached some remarks on the floor function and a similar inequality that is true. I think it's worth your while to read at least the easy facts about floor.
Wow, thank you very much for the two informative sheets!
It helped that the facts came with proof.
I think I was able to become a bit more familiar with the floor function and I managed to follow through your last proof.