Is this 'floor' proof correct.

For show that .

My proof...

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 .

Then .

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.