Hi I am working my way through this interval analysis textbook but I am confused about one of the properties (Theorem 6.1 on page 55-56)
http://interval.ict.nsc.ru/Library/InteBooks/IntroIntervAnal-2009.pdf
It states the following:
If $F(X)$ is an inclusion isotonic, Lipschitz, interval extension then the excess width of a refinement $F_{(N)}(X)$ is of order 1/N.
I understand the following, so that $F_{(N)}(X) = f(X_{1},...,X_{n}) + E_(N)$ where the $w(E_N) \leq K w(X)/N$ where $N$ is the number of subdivisions and $K$ is the Lipschitz constant.
I also understand the proof of the following: $w(E_s ) = w(F (X_s )) − w(f (X_s )) ≤ w(F (X_s )) ≤ Lw(X_s ) ≤ Lw(X)/N$ for each $X_s$
However what I do not understand is this part:
The inequality for $w(E_N)$ holds with K = 2L since, in the worst case, the maximum excess width may have to be added to both upper and lower bounds in the union.
Why is it 2 * L and not multiplied by the number of subdivisions (e.g. the sum of total excess width of each subdivision).
I understand that this is likely because each of the subdivisions overlap and therefore we can ignore the excess width of each division except the two at the ends of the list, but I cannot seem to find a proof to clarify this. If someone could explain this in more detail (or provide a step by step proof) it would be very appreciated.
Thank you :)