Editing Heine–Borel theorem (section)

==History and motivation==

The history of what today is called the Heine–Borel theorem starts in the 19th century, with the search for solid foundations of real analysis. Central to the theory was the concept of [[uniform continuity]] and the theorem stating that every [[continuous function]] on a closed and bounded interval is uniformly continuous. [[Peter Gustav Lejeune Dirichlet]] was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof.<ref name="Sundström"/> He used this proof in his 1852 lectures, which were published only in 1904.<ref name="Sundström">{{cite journal | journal = [[American Mathematical Monthly]] | title = A Pedagogical History of Compactness | last1 = Raman-Sundström | first1 = Manya | date = August–September 2015 | volume = 122 | issue = 7 | pages = 619–635 | jstor = 10.4169/amer.math.monthly.122.7.619| doi = 10.4169/amer.math.monthly.122.7.619 | arxiv = 1006.4131 | s2cid = 119936587 }}</ref> Later [[Eduard Heine]], [[Karl Weierstrass]] and [[Salvatore Pincherle]] used similar techniques. [[Émile Borel]] in 1895 was the first to state and prove a form of what is now called the Heine–Borel theorem. His formulation was restricted to [[countable set|countable]] covers. Pierre Cousin (1895), [[Lebesgue]] (1898) and [[Arthur Schoenflies|Schoenflies]] (1900) generalized it to arbitrary covers.<ref name="sundstrom_2010">{{cite arXiv |last=Sundström |first=Manya Raman | eprint=1006.4131v1 |title=A pedagogical history of compactness |class=math.HO |year=2010 }}</ref>