I have noticed that two of the math books used in high school goes about the binomial theorem in this way:
- Prove it on integers using induction
- Generalize it and use it in proofs needed to develop calculus
- Use calculus to prove the binomial theorem for $\mathbb{R}$
In college level text books (Rudin...) this approach is of cause not taken, but these are often to advanced to show in high school.
There is a rather simple outline here that develops most of the foundation needed for proving the binomial theorem for $\mathbb{R}$. But I was wondering if anyone knows of a simple proof suitable to showing in a high school class for interested although not advanced students ?.