I want to prove that the NP-hardness of Maximum Independent Set implies that there is no FPTAS for the Maximum Independent Set problem unless $P=NP$.
I found the following approach after some research online:
Assuming that such an FPTAS exists which computes a $(1 - \epsilon)$-approximation for Maximum Independent Set on a graph G. I want to use this assumed algorithm as a subroutine in a new algorithm to solve the maximum independent set problem exactly. Now proving that this new algorithm runs in polynomial time would be a contradiction (assuming $P \neq NP)$ since this would mean that I solved the maximum independent set problem in polynomial time. Therefore there exists no FPTAS.
I can't figure out a polynomial time algorithm which uses the above assumed subroutine. How does this algorithm look like, how to use the assumed subroutine (for example what $\epsilon$ to choose?).