Suppose Q(a) = 0 and the root a of this equation has multiplicity k.

Then we have Q(x) = (x - a)^kZ(x) and Z(a) = c for non-zero c.

Suppose further that R(a) = d for non-zero d. (otherwise we could factor (x - a) out of both R and Q and reduce their degrees.)

Then behaves at worst like  at x = a, because the rational function vanishes at x = a and must therefore have a factor (x - a) in it.

This means that by substraction of an appropriate multiple of an appropriate inverse power, we can obtain a rational function that is less singular than R(x) / Q(x) at an arbitrary root of Q

If we continue such substractions until we have removed all the singularities of R(x) / Q(x) at all the roots, we will be left with a rational function that still vanishes at infinity, and now has no finite singularities. The only such function is 0; so that R(x) / Q(x) must equal the sum of the substractions; which statement is the theorem.

Comment: proof details