Of course free-will exists, can there be any doubt?
The problem is that if there cannot be any doubt then we are not free to doubt its existence. And of course if there is doubt, it is possible that free-will does not exist at all.
This paradox, is surely due to the way our minds think and are arranged to think. A less logical system for performing these thinking tasks might be able (for instance) to contain two opposing ideas simultaneously. Such a system would not be useful with our present most common kind of thinking logic, but it could possibly work when one replaces exact definitions with double definitions of the same thing in two different ways.
We are close to this in problems in macroeconomics when we have to assume as twin axioms that: Man seeks to satisfy his very many desires, but with the least amount of effort. (Originally from Henry George, 1879, "Progress and Poverty".) This oxymoron situation clearly leads to our having to consider the macroeconomics system of containing a combination of two opposites at every place.
So it seems to me that in trying to prove the existence of free-will, we are asking what is actually an illegitimate question.