At the end of season 5:
We see the Night King raise all the dead soldiers of Hardholme into wights. This seems to be done through the Night Kings proximity to the dead soliders.
In the most recent episode of season 7:
We see one of the dragons turn into a wight dragon. I assume this was done by the Night King felling the dragon with his spear.
However, in season 1:
The corpses of two members of Benjen Stark's scouting party are found near the Wall. They are taken back through the Wall to Castle Black where they eventually reanimate as wights, and they attack Jon Snow and Lord Commander Jeor Mormont.
So my main question is, how are wights reanimated?
The events in season 1 seems to contradict the idea of the Night King needing to be in proximity to turn the dead into wights. If wights can be reanimated beyond the wall why doesnt the Night King reanimate all the dead he can beyond the wall? Or was something done to those two corpses (perhaps in the way they were killed) that made them eventually turn into wights?