A more intuitive proof of the Zone theorem ?
This is not cleaner, but it is a good preparation for more advanced stuff, and it is a good example of abstraction...

One can use Davenport-Schinzel sequences argument. Consider the region above your zone line. Every line becomes a ray, and in fact two rays, as we consider the left side and right side as being different. Scan the boundary of this zone from left to right, writing down which rays you encounter. This is a sequence defined over 2n symbols, and the pattern abab is illegal. As such, the length of the sequence is at most 2(2n)-1 = 4n-1. Applying it to the zone below the line, implies a bound of the form 8n.

Now, proving that a sequence of symbols without ...a..b..a..b... as a subsequence of n symbols has length 2n-1 is easy. indeed, consider two consecutive appearances of the same character that are closest to each other in this sequence. Clearly, in between these two characters, each character that appears must be unique. Consider such a character, and observe that if it appears anywhere else in the string, then we will get the forbidden subsequence. As such, this character appears exactly once in the string. Remove it, and remove an extra character if needed if you created two consecutive identical characters. Namely, removing a character from the string shorten it by 2, as such, the maximum length of the string is 2n-1.