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.