Title: The Dual Sovereign Doctrine and Double Jeopardy: Examining the Legality of Prosecution in Both State and Federal Courts

The text notes that we have a dual court system (state and federal courts). The Fifth Amendment of the U.S. Constitution guarantees, among other things, that no “person be subject for the same offense to be twice put in jeopardy of life or limb” (no double jeopardy).  
However, the separate/dual sovereign doctrine has been applied to allow a state and the federal government to both bring charges against an individual for the same crime.
See the video below on Gamble, an Alabama case.  
Discuss the separate/dual sovereign doctrine. Should prosecution for a crime in both state and federal courts be prohibited by the double jeopardy clause? Fully explain your answer.