Sequential compactness

Abstract

Abstract We recall the form of Bolzano–Weierstrass theorem stated in Chapter 4: any bounded sequence of real numbers has at least one convergent subsequence. (You may have seen an equivalent form, which says that any bounded infinite subset of real numbers has at least one limit point; the sequence version is more relevant to our purposes.) The Bolzano– Weierstrass theorem is closely related to the one-dimensional Heine Borel theorem. For example, we have proved that a continuous real-valued function on a closed bounded interval [a, b] is bounded and attains its bounds using the Heine–Borel theorem (more precisely, the special case for [a, b]); equally well this can be proved using the Bolzano–theorem— in analysis textbooks it is often proved that way, for example see Theorems 4.3.1 and 4.3.2 of Hart (2001).