We write the problem as following: Consider a finite set of closed intervals on a line, such that for any $k$ of them $2$ intersect. Show we can place $k-1$ points such that every interval contains at least one of those. We proceed by induction on $k$. For $k=2$ it is Helly's theorem. Suppose it is true for $k-1$,we will show it is true for $k$. Take the interval $I_n=[a_n,b_n]$ with maximal $a_n$, and place a point at $a_n$. Now, remove all intervals that have a common point with it, and consider the remaining ones. We need to show that for any $k-2$ of them $2$ intersect, and we will be done by induction. Suppose that we have $k-1$ intervals pairwise-non intersecting. Then, none of them intersects $I_n$ so we get a set of $k$ intervals pairwise non-intersecting,contradiction!