proving fractional Helly's theorem for boxes and rectangles

Question

I'm having a hard time proving the followings. the background: I'm self-learning combinatorial geometry based on 'lectures on Discrete Geometry' by Matousek:

Q1.Let B be a finite family of axis-parallel boxes in $R^{d}$ such that any two have a non empty intersection. prove that they all have a non empty intersection.

Q2. Let R be a finite collection of axis parallel rectangles in $R^{2}$ such that out of any three, there are some two that have a non-empty intersection.Prove that the rectangles in R can be stabbed by 3 points and show that the number 3 is the best here.

I know the proofs are based on Helly's Theorem\ Fractional Helly's Theorem , but wasn't able to make progress.

Any proofs, hints, leads , thoughts would be much appreciated !

Answer 1

Question $1$ is quite easy. Just project the family of boxes onto each of the $d$ dimensions & apply Helly's theorem for each of these $1$ dimensional cases. Then the product of each of these $1$ dimensional intersections will intersect each of the boxes.

Not sure how to do Q$2$.