Proof of Newton's Generalized Binomial Theorem (without Calculus)

Question

I'm writing an article for derivates, I've already prooved Newton's Binomial Theorem, but I want to proof that the expresion $$(a+b)^r=\sum_{i=0}^\infty\binom{r}{i}a^ib^{r-i}$$ works for all $a,b,r\in\mathbb{R}$, where $$\binom{r}{i}:=\frac{r(r-1)\cdots(r-(i-1))}{i!}$$