Let be p(x)=a3*x^3 + a2*x^2 + a1*x + a0 so that O((x-1)^4)=p(x) at x=1 neighborhood.
Prove: p(x)=0 for every x.
is a cubic in and that at
This means that there exists a such that:
Divide through and we have:
But for small enough we have:
So we have:
but as the left hand side goes to as goes to this is a contradiction unless .
This construction can now be repeated to show that b , which means we have proven that , and so