| p
|
|
| | p + r
|
| (≥ 0) | | r
|
| (≥ 0) | | -p
|
| (≤ 0) | | p - r
|
| (≥ 0) | | 2∙p - r
|
| (≥ 0) | | p
|
| (≥ 0) | | r - p
|
| (≤ 0) | | - r
|
| (≤ 0) | | p
|
| (≥ 0) | | p + r
|
|
|
| From (0) we conclude (i) that the sequence contains a nonnegative element, (ii) that one of its neighbours is nonnegative, and (iii) that at least one of the two elements adjacent to a pair of nonnegative neighbours is nonnegative. More precisely: the sequence contains in some direction a triple of adjacent elements of the form (p, p+r, r) with 0 ≤ r ≤ p. To the left we have extended the sequence with another 8 elements. From (0) we further conclude that the whole sequence is determined by a pair of adjacent values; hence, the repetition of the pair (p, p+r) at distance 9 proves the theorem. [The above deserves recording for its lack of case analyses.] |