How was Copernicus proven correct, and what were the implications for those who validated his theories?
Because the study of astronomy was closely tied to the theological doctrine of the Church during the 16th Century, the time in which Nicholaus Copernicus (1473-1543) was active, any academic theory that deviated from that doctrine could prove very injurious to the individual propagating that theory. Copernicus not only questioned the Church-sanctioned theory of an Earth-centric universe, the shattered it with mathematical models. While Copernicus' theory of a heliocentric universe (or solar system) was radical, however, he was spared the wrath of the Church and died of a stroke at the age of 70.
While Copernicus contributed major advances in the study of astronomy by virtue of his mathematical models supporting the notion of a Sun-centered solar system, physics had not yet progressed to the level necessary to indisputably validate his theory. It would be Galileo, with his use of telescopes, who would make major contributions in support of Copernicus' heliocentric theory of the solar system, and who also risked his life in the interest of scientific discovery, the Inquisition being a serious threat to those who contradicted the teachings of the Church. He wasn't executed, but was kept under house arrest for the remainder of his life.
Other prominent scientists who contributed greatly to the study of astronomy during the same period included Johannes Kepler and Tycho Brahe. Kepler in particular was noted for his contributions to the study of astronomy while also holding fast to his conviction of a universe created by a Supreme Being.