How did Americans obtain Sioux land?

Beginning in 1492 Diseases Brought from Europe and Africa reduced the native American population from 13 million to 2 million then genocide reduced them to less than 1 million. In the 300 years of depopulation and slaughter of the buffalo, elk and deer it made room for Europeans also giving them a numbers advantage to win wars against weakened Natives. Gold was found were treaties had awarded land to Sioux's so the military broke the treaty to protect the large number of non Indians searching for gold. US government made promises they didn't intend to keep so encroachment was Status Quot. The 1868 treaty was less than a year old before the USA had broken promises to protect Natives from whites forcing Chiefs to listen to the young warriors, who were in many cases starving, to go to war which they eventually lost even though winning some outstanding battles against repeating rifles with lances, Bows and a few muzzle loaders from the backs of horses. Wounded Knee wasn't a battle or war it was pure and simple genocide. After blaming Indians for their deficiencies they reduce the size of reservations as often as they felt entitled to do so.