Editing Subsequence (section)

== Theorems ==

* Every infinite sequence of [[real number]]s has an infinite [[Monotone sequence|monotone]] subsequence (This is a lemma used in the [[Bolzano–Weierstrass theorem#Proof|proof of the Bolzano–Weierstrass theorem]]).
* Every infinite [[bounded sequence]] in <math>\R^n</math> has a [[Limit of a sequence|convergent]] subsequence (This is the [[Bolzano–Weierstrass theorem]]).
* For all [[integer]]s <math>r</math> and <math>s,</math> every finite sequence of length at least <math>(r - 1)(s - 1) + 1</math> contains a monotonically increasing subsequence of length&nbsp;<math>r</math> {{em|or}} a monotonically decreasing subsequence of length&nbsp;<math>s</math> (This is the [[Erdős–Szekeres theorem]]).
* A metric space <math>(X,d)</math> is compact if every sequence in <math>X</math> has a convergent subsequence whose limit is in <math>X</math>.