Is this 'floor' proof correct.
For show that .
If both sides will be 0 so we will deal with the case when .
Let be the largest integer smaller than such that .
Then we have where and .
So and since ,
Now let where .
And since , , hence .
So . Hence the two are equal.
I hope I've written this out right. To explain if it looks wrong... Imagine, 13.86/4. Then the largest integer smaller than that divides would be 12, would be 1 and would be 0.86.
EDIT: I suppose this proof would also work for the case for actually.