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:
“The process of writing has something infinite about it. Even though it is interrupted each night, it is one single notation.”
—Elias Canetti (b. 1905)
“I wonder whether mankind could not get along without all these names, which keep increasing every day, and hour, and moment; till at the last the very air will be full of them; and even in a great plain, men will be breathing each other’s breath, owing to the vast multitude of words they use, that consume all the air, just as lamp-burners do gas.”
—Herman Melville (1819–1891)