You are correct. No point on the plane projects to the North pole itself. But as a sequence of points move out towards infinity on the plane, their projections move towards the North pole on the sphere. So you now declare that infinity is a point, though not a point on the plane. It's an additional point. Its projection is the North pole of the sphere.
You are correct. No point on the plane projects to the North pole itself. But as a sequence of points move out towards infinity on the plane, their projections move towards the North pole on the sphere. So you now declare that infinity is a point, though not a point on the plane. It's an additional point. Its projection is the North pole of the sphere.