De Bruijn--Erdos pair-covering inequality

If $r_x$ denotes the number of lines through $x$ in a nontrivial finite linear space on $v$ points, thenTo prove it, let $k_L$ be the line sizes. The inequalities $k_L\le r_x$ whenever $x\notin L$, counted at each integer threshold, givefor every $t\ge 1$. Summing in $t$ and using $\left(\genfrac{}{}{0ex}{}{s}{2}\right)={\sum }_{t\ge 1}(s-t{)}_{+}$ giveswhere the last identity counts pairs of points by their unique line.