I try the method of induction on it, but I fail at the last step. I assume that the statement is true for all planes with less than n points. Then if I add one more point to the plane so that it is not collinear to the line with exaclty two points on it, the statement is true for the plane with n points. However, if the new point is added on the line with exactly two points, how can we make sure that there is still a line passing through exactly two points?
This is called the Sylvester-Gallai Theorem. You can find many proofs on the internet, including in the Wikipedia article:
See the following notes for a nice and slick proof: http://web.stanford.edu/~yuvalwig/math/teaching/WhatsThePoint.pdf