Using general relativity (which has plenty of equations), and a little borrowed knowledge from other fields (to describe star collapse), you can show that black holes should exist. But unfortunately there are no proofs in physics, just experimental and observational evidence.
That being said, the observational evidence of the existence of the black holes has been extremely good. For example, by looking at the movement of the stars in the galactic core we’ve determined that they must be orbiting a tiny, invisible object with several million Suns worth of mass.
Which sounds like pretty good evidence for a black hole! Of course, at the end of every theory, proposition, and paper is a tiny invisible asterisk that reads:
“*All of the above assumes that something we’ve never heard of and/or could never have imagined isn’t what’s actually going on. That would hella suck.”
From time to time “something we’ve never heard of” is exactly what’s going on, but there isn’t a lot that can be done about that. Dismissing a working theory because something might be wrong is paralyzing.
So Sagittarius A* (the super-massive object in the center of our galaxy) is almost definitely a black hole, but it hasn’t been (and never will be completely, absolutely, and totally) proven that black holes exist. I’m convinced though.