Strayer Elementary Number Theory 2. 1. 27.

 

Just a note: I refer to Ex. 15.c. in my proof. If you actually look at Ex. 15.c. in the book, for some strange reason it prohibits that a=0. I don't understand why. Of course it is imperative to not allow b=0 in this proof or any similar theory/ proofs (like the Division Algorithm) because of division by zero reasons. But a=0 is fine. The theory works even if a=0, you just won't get any interesting results. So in the proof of Ex. 15.c. I uploaded to this website at an earlier date, I allow a=0 and the proof works just fine. If you're confused as to what I'm talking about, go check it out.

Anyways allowing a=0 lets me reference Ex.15.c. below and use it in my proof FOR ALL integers, not just non-zero integers. 

The 'Lemma' in the proof references something I've since deleted but really it was just a proof that ⌈x+y⌉=⌈x⌉+y and ⌊x+y⌋=⌊x⌋+y for real number x and integer y; a proof of the latter is on my blog here.