In one sense, this is a
definition, not a theorem that needs a proof. We define the remainder as having a degree less than the divisor.
What needs proof is the existence and uniqueness of the whole thing -- that for any polynomials a and b, polynomials q and r satisfying the definition
exist, and there are
only one of each. You can find proofs all over the place, for example
here and
here. But in a sense, the proof is seen whenever you carry out such a division -- the fact that there is a way to do it tells you that it can be done (and, in particular, that you can get the degree of the remainder to be less than that of the divisor).
The statement about the degrees of q, a, and b is implied by the definition, since when you multiply b and q, the degree of the product, a, is the sum of the degrees of b and q.
