Monotone Subsequence Lemma
Every real sequence has a monotone subsequence.
Monotone subsequence lemma: Every real sequence has a subsequence that is monotone (either nondecreasing or nonincreasing).
A sequence in a set is a function , where is the natural numbers. One writes for , so the sequence is denoted .
A subsequence of a sequence is a sequence of the form , where is a strictly increasing sequence of indices.
Subsequences are a standard way to isolate “partial” limiting behavior of a sequence; they are used in defining limit superior and limit inferior. The monotone subsequence lemma guarantees monotone subsequences under mild hypotheses.
A monotone sequence is a sequence of real numbers such that either
- for all (monotone increasing), or
- for all (monotone decreasing).
A limit superior of a real sequence is the extended real number
provided the right-hand side is interpreted in .
The limit inferior of a sequence of extended real numbers is
where the supremum and infima are taken in .