Question
Why is the following statement true?
Given a $N \times N$ grid and any space-filling curve $f: N \times N \rightarrow \{ 1,...,N^2 \}$ that covers the entire grid, the cells on the grid which are assigned the numbers $1$ and $N^2$ (the "first" and "last" numbers) have $L_\infty$-distance of at most $N-1$, and therefore there must exist two cells which are direct neighbors on the grid but have a distance of at least $N+1$ on the curve.
Background
The previous statement is my slightly paraphrased version of a short assertion/statement made in a paper about space-filling curves I am studying. The first part to me is clear:
- The cells assigned numbers $1$ and $N^2$ are at a distance of at most $N-1$ on the grid.
(Using Chebyshev/chessboard distance this is the farthest away two cells can be on the grid). But I do not see how it implies the second statement:
- There always exist two directly neighboring cells whose numbers differ by at least $N+1$.
The statement is worded such that the conclusion is obvious, but I must admit it is not obvious to me. The statement in question is in the very last paragraph of page labeled 8 in the full PDF available here. I will also note that for the first part of the statement the author uses the language "there is a path of length at most $N-1$" which seems strange to me and I cannot think of another way to interpret it besides the one I have stated.