One more proof of Corollary 4.3 is given in "Common Factors of Integers: a Graphic View" by R.B. Eggleton, Discrete Mathematics, 65 (2) 1987, pp. 141-147.
In the author's opinion, this is the most beautiful proof she has seen so far that stapled sequences of length N exist for every N > 16. The author is grateful to Prof. Yair Caro for providing this reference.