How did humans discover black holes? How far is the black hole from the earth?

It was originally a solution and theory calculated by Schwarzschild according to general relativity. Later observation confirmed it.

How far is it? . . Sorry, there are many black holes in the universe. The nearest black hole is located in Sagittarius, which forms a binary system with an ordinary star numbered V464 1SGR, about 1600 light-years away from the Earth.