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Heine-Borel theorem

The Heine-Borel Theorem in analysis states:
A subset of the real numbers R is compact iff it is closed and bounded.

The central idea arose from trying to find uniform bounds on the behavior of a function over all points in a set. Such bounds could often be found for some small open interval about any point in a set, but the question became, could these bounds somehow be combined to form global uniform bounds over the entire set? This led to open covers and the concept of compactness, which led to the actual theorem. The Theorem of Bolzano-Weierstrass is closely related.

The theorem is true not only for the real numbers, but also for some other metric spaces: the complex numbers, the p-adic numbers, and Euclidean space Rn. However, it fails for the rational numbers and for infinite dimensional normed vector spaces. The proper generalization to arbitrary metric spaces is:

A subset of a metric space is compact if and only if it is complete and totally bounded.

Here is a sketch of the "=>"-part of the proof according to Jean Dieudonné:
  1. It is obvious that any compact set E is totally bounded.
  2. Let (xn) be an arbitrary Cauchy sequence in E; let Fn be the closure of the set {xk : k >= n} in E and Un := E - Fn. If the intersection of all Fn would be empty, (Un) would be an open cover of E, hence there would be a finite subcover (Unk) of E, hence the intersection of the Fnk would be empty; this implies that Fn is empty for all n larger than any of the nk, which is a contradiction. Hence, the intersection of all Fn is not empty, and any point in this intersection is an acculumation point of the sequence (xn).
  3. Any accumulation point of a Cauchy sequence is a limit point (xn); hence any Cauchy sequence in E converges in E, in other words: E is complete.

A proof of the "<="-part can be sketched as follows:
  1. If E would not be compact, there would exist a cover (Ul)l of E having no finite subcover of E. Use the total boundedness of E to define inductively a sequence of balls (Bn) in E with
    • the radius of Bn is 2-n;
    • there is no finite subcover (Ul∩Bn)l of Bn;
    • Bn+1∩Bn is not empty.
  2. Let xn be the center point of Bn and let yn be any point in Bn+1∩Bn; hence we have d(xn+1,xn) <= d(xn+1,yn)+d(yn,xn) <= 2-n-1+2-n <= 2-n+1. It follows for n <= p < q: d(xp,xq) <= d(xp,xp+1) + ... + d(xq-1,xq) <= 2-p+1 + ... + 2-q+2 <= 2-n+2. Therefore, (xn) is a Cauchy sequence in E, converging to some limit point a in E, because E is complete.
  3. Let l0 be an index such that Ul0 contains a; since (xn) converges to a and Ul0 is open, there is a large n such that the ball Bn is a subset of Ul0 - in contradiction to the construction of Bn.