The Heine-Borel theorem states that a subspace of 
(with the usual topology) is compact iff it is closed and bounded.
The Heine-Borel theorem can be proved using the Bolzano-Weierstrass theorem.
Heine-Borel Theorem
See also
Bolzano-Weierstrass Theorem, Bounded Set, Compact SpaceExplore with Wolfram|Alpha
References
Baker, H. F. Cited in Lamb, H. Proc. London Math. Soc. 35, 459-460, 1903.Heine, E. "Die Elemente der Functionenlehre." J. reine angew. Math. 74, 172-188, 1871.Jeffreys, H. and Jeffreys, B. S. "The Heine-Borel Theorem" and "The Modified Heine-Borel Theorem." §1.0621-1.0622 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 20-21, 1988.Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 9, 1996.Young, W. H. "Overlapping Intervals." Proc. London Math. Soc. 35, 384-388, 1903.Referenced on Wolfram|Alpha
Heine-Borel TheoremCite this as:
Weisstein, Eric W. "Heine-Borel Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Heine-BorelTheorem.html