Let $J$ be a family of parallel closed intervals in the plane $\mathbb{R}^2$, no two different of them contained in any common line. Moreover, for arbitrary three intervals belonging to $J$ there exists a line intersecting all three of them. I would like to prove that there exists a line intersecting all segments belonging to $J$! (I'd like to add that $J$ can be infinite).
  
  

It seems to be rather complicated even in the case of $4$ lines! Moreover, the statement is wrong if we replace segments by open intervals: there is no line intersecting each interval of the family $\{(0,\frac{1}{n})\times \{n\}: n\in \mathbb{N}\}$, but each triple of course can be intersected by some vertical line.  
  

Can anybody please suggest something?
  
  

I was trying to prove some simple finite cases but failed. I think it is important to neglect specificity of finiteness: the set of points of segments can even be dense!



Segments on a family of parallel lines
Here is a proof for a finite number of parallel lines. For each segment $s\_i$, let $L\_i$ be the set of lines that intersect $s\_i$. We may regard each line $\ell \in L\_i$ as a point $(m,b) \in \mathbb{R}^2$, where $m$ is the slope of $\ell$ and $b$ is its $y$-intercept. In this way, we may regard $L\_i$ as a subset of $\mathbb{R}^2$. It is easy to check that $L\_i$ is convex for all $i \in I$. By hypothesis, $L\_i \cap L\_j \cap L\_k$ is non-empty for all $i,j,k$. Thus, by [Helly's theorem](http://en.wikipedia.org/wiki/Helly%27s_theorem), $\bigcap\_{i \in I} L\_i$ is non-empty. That is, there is a line that passes through each $s\_i$. 


There is a version of Helly's theorem which applies to infinite families of convex sets, but with the additional requirement that the sets also be compact. Unfortunately for us, while $L\_i$ is convex, it is not compact because the $y$-intercepts always take on all values in $(-\infty, \infty)$. 


**Edit.** There is a suitable version of Helly's theorem for infinite families which implies the result for infinite $I$. You can find the version [here](http://users.mat.unimi.it/users/libor/AnConvessa/Helly_Thm.pdf), together with a proof of your result, which is referred to as the *Common Transversal Theorem* or the *Skewer Theorem*.


