I just read a very short paper on the De Bruijn-Erdos theorem. It essentially states that if we have $n$ points and $m$ lines, where any two points lie on a line, and each line has at least two points, then there are at least as many lines as points.
The proof is in this short paper. It is stated at the very top, and the proof I'm curious about begins in the middle of page 2 and ends at the top of page 3. I don't quite follow the inequalities $ s_j\leq k_n\ \text{for}\ j>\nu $ that are found at the very top of page 3. Why would that fact that an index $j$ being greater than the minimum number of lines passing through a given point require that the number of points on that indexed line is at most the number of lines passing through that point? Apart from that I follow the rest. Does it violate one of the assumptions about the space? Thanks.