Infinite Increasing Subsequences
If (, ≤) is wqo then every infinite sequence, … contains an infinite increasing subsequence ≤≤≤… (with <<<…). Such a subsequence is sometimes called perfect. This can be proved by a Ramsey argument: given some sequence, consider the set of indexes such that has no larger or equal to its right, i.e., with . If is infinite, then the -extracted subsequence contradicts the assumption that is wqo. So is finite, and any with larger than any index in can be used as the starting point of an infinite increasing subsequence.
The existence of such infinite increasing subsequences is sometimes taken as a definition for well-quasi-ordering, leading to an equivalent notion.
Read more about this topic: Well-quasi-ordering
Famous quotes containing the words infinite and/or increasing:
“It has no future but itself—
Its infinite contain
Its past—enlightened to perceive
New periods of pain.”
—Emily Dickinson (1830–1886)
“Yet I doubt not through the ages one increasing purpose runs,
And the thoughts of men are widened with the process of the suns.”
—Alfred Tennyson (1809–1892)