let

be the n roots of a polynomial

Note: The derivative has degree n-1 so has at most n-1 roots

Consider the following by Rolle's theorem.

since

for i =1 to n-1

Then for each i There exists a

such that

These are each of the n-1 zeros of the derivative.

I guess this proof only works the zero's are not repeated(I think)

QED

I hope this helps

Good luck