Born in 1927, Finesilver graduated from the University of Denver School of Law in 1952. He served as an assistant district attorney for the city of Denver from 1952-1955, and was later appointed to the Federal Judiciary by President Richard Nixon in 1971. From 1982-1994 he served as chief judge on the Federal Judiciary. He died in Denver in 2006.
Finesilver held the following positions throughout his career:
Assistant Denver City & County Attorney, Denver, 1952-55
Municipal Judge, Denver, 1955-1962
Denver District Judge, 1962-71
US 10th Circuit Judge, 1971-94