The “Mountain” title is an historical title only. They are not mountain troops, but Light infantry, capable of fighting in any environment. Just like the British call some of their tank units “Lancers”. They haven’t seen a horse or a lance in years.
The 10th Mountain Division originated at Camp Hale The history of Camp Hale Camp Hale is located between 10.000 feet and 14.000 feet in altitude. Considerably better for training a Mountain Division than at sea level. You seem woefully ignorant of history.