With his new tool, the telescope, Galileo discovered that Jupiter had four mysterious bodies moving around it. He theorized they must be moons, much as ours. This discovery was particularly important because it showed that there could be centers of motion that in turn are in motion. It had previously been argued that if the Earth were going around the Sun, the Moon would be left far behind because it could hardly keep up with the rapidly moving planet! Yet here were Jupiter's satellites doing exactly that.
But this was not the most convincing argument used to prove the Earth goes around the Sun. Try another answer.
(By the way, why doesn't the Earth's Moon get left behind?)