We have this interesting problem with black holes. What is a black hole? It is a region of space where you have mass that's confined to zero volume, which means that the density is infinitely large, which means we have no way of describing, really, what a black hole is!
Proof of the black hole is a tremendous amount of mass inside a very small volume. There's 4 million times the mass of our sun within a region that's comparable to the size of our solar system.
The key to proving that there's a black hole is showing that there's a tremendous amount of mass in a very small volume. And you can do that with the motions of stars.