How did we realize that light travels at a finite speed?

As far as I know the first time anyone proved that light has a finite speed was when the astronomer Rømer discovered variations in the timings in the transits of Jupiter's moons. He correctly attributed this to the time light took to reach Earth from Jupiter.

His calculated value for $c$ was about 26% too low, but that was pretty good given the state of the art at that time.