From Wikipedia: "There is no universally-accepted definition of mountain. In the United States, the following points of measurement have been used and taught in geography classes:
* Flat to 500 feet, base to highest point - Rolling Plain
* Highest point 501 to 999 feet above base - Hill
* Highest point 1000 feet or more above base - Mountain"
There was an excellent film released in 1995 entitled "The Englishman Who Went Up a Hill But Came Down a Mountain":
"The movie is set in 1917 (with World War I in the background) and revolves around two English cartographers, the pompous Garrad and his junior, Anson, who arrive at the fictional Welsh village of Ffynnon Garw ("Rough Fountain" in Welsh) to measure its "mountain" - only to cause outrage when they conclude that it is only a hill because it is slightly short of the required 1000 feet."
https://en.wikipedia.org/wiki/The_Englishman_Who_Went_Up_a_Hill_But_Came_Down_a_Mountain
Again, note that a mountain starts at 1000ft. Therefore, any state whose highest point is less than 1000ft has no mountains, despite the whims of the publicists or wishful thinking. The defence rests.
This site lists those highest points:
http://geology.com/state-high-points.shtml
Thus, based on the definition of "1000ft = a mountain", we can conclude that the states with no mountains =
Delaware
Florida
Louisiana
Mississippi
Rhode Island
Using dictionary.com's definition of "2000ft = a mountain", we get states with no mountains =
Delaware
Florida
Louisiana
Mississippi
Rhode Island
Illinois
Indiana
Iowa
Michigan
Missouri
New Jersey
Ohio
Wisconsin