Editing Bolzano–Weierstrass theorem (section)

== Sequential compactness in Euclidean spaces ==

'''Definition:''' A set ''<math>A \subseteq \mathbb{R}^n</math>'' is [[sequentially compact]] if every sequence ''<math>\{x_n\}</math>'' in ''<math>A</math>'' has a convergent subsequence converging to an element of ''<math>A</math>''.

'''Theorem:'''  ''<math>A \subseteq \mathbb{R}^n</math>'' is [[sequentially compact]] if and only if ''<math>A</math>'' is [[Closed set|closed]] and bounded.

'''Proof:''' ([[sequentially compact|sequential compactness]] implies closed and bounded)

Suppose ''<math>A</math>'' is a subset of <math>\R^n</math> with the property that every sequence in ''<math>A</math>'' has a subsequence converging to an element of ''<math>A</math>''. Then ''<math>A</math>'' must be bounded, since otherwise the following unbounded sequence <math>\{x_n \} \in A</math> can be constructed. For every <math>n \in \mathbb{N}</math>, define <math>x_n</math> to be any arbitrary point such that <math>|| x_n || \geq n</math>. Then, every subsequence of <math>\{x_n\}</math> is unbounded and therefore not convergent. Moreover, ''<math>A</math>'' must be closed, since any [[Accumulation point|limit point]] of ''<math>A</math>'', which has a sequence of points in ''<math>A</math>'' converging to itself, must also lie in ''<math>A</math>.''

'''Proof:''' (closed and bounded implies [[sequentially compact|sequential compactness]])

Since ''<math>A</math>'' is bounded, any sequence ''<math>\{x_n\}\in A</math>'' is also bounded. From the [[Bolzano-Weierstrass theorem]], ''<math>\{x_n\}</math>'' contains a subsequence converging to some point <math>x \in\R^n</math>. Since <math>x</math> is a [[Accumulation point|limit point]] of ''<math>A</math>'' and ''<math>A</math>'' is a [[closed set]], ''<math>x</math>'' must be an element of ''<math>A</math>''.

Thus the subsets ''<math>A</math>'' of <math>\R^n</math> for which every sequence in ''A'' has a subsequence converging to an element of ''<math>A</math>'' &ndash; i.e., the subsets that are [[sequentially compact]] in the [[subspace topology]]&nbsp;&ndash; are precisely the closed and bounded subsets.

This form of the theorem makes especially clear the analogy to the [[Heine–Borel theorem]], which asserts that a subset of <math>\R^n</math> is [[Compact space|compact]] if and only if it is closed and bounded.  In fact, general topology tells us that a [[metrizable space]] is compact if and only if it is sequentially compact, so that the Bolzano–Weierstrass and Heine–Borel theorems are essentially the same.