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.
I'm admittedly way out of my league here, especially when constructing proofs, but why is all that rigor necessary?
1) It seems self-evident that where is a factor of , also meaning , then and .
2) For , is a factor of .
3) Since , then .
Sorry, I'm not at all a mathematician, so I'm not certain how to express this properly, let alone elegantly. It seems to me that it should be satisfactory to prove it for two simple and mutually-exclusive cases: and , as described below...
Let . Where , is an integer, therefore and .
Let and . This means . In other words, because , can not be less than .
is an integer only when . Therefore, . Where , .
represents , so .
If formalized, could this be a reasonable approach to proving for and ?