“and I gave Mount Seir to Esau.” Mount Seir is the biblical name for the mountainous area stretching between the south end of the Dead Sea and the north end of the Gulf of Aqabah. It is not all of biblical Edom, but the northwest part of it. It was originally controlled by a Horite name Seir (Gen. 14:6; 36:20), but Esau and his descendants conquered it and took it from the Horites and then lived in the area (Deut. 2:12, 22).
“but Jacob and his children went down into Egypt.” It is interesting that God gave Esau the area of Mount Seir, Edom, and Esau became a nation there. But the spiritual battle around the Promised Land was such that although Jacob had been given the land of Israel, the Israelites did not become a nation there, but were made into a nation while they were slaves in Egypt, and came out and traveled and fought as one nation although composed of different tribes.