Theorem of Heine-Borel

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 (x_n) be an arbitrary Cauchy sequence in E; let F_n be the closure of the set {x_k : k >= n} in E and U_n := E - F_n. If the intersection of all F_n would be empty, (U_n) would be an open cover of E, hence there would be a finite subcover (U_nk) of E, hence the intersection of the F_nk would be empty; this implies that F_n is empty for all n larger than any of the n_k, which is a contradiction. Hence, the intersection of all F_n is not empty, and any point in this intersection is an acculumation point of the sequence (x_n).
3. Any accumulation point of a Cauchy sequence is a limit point (x_n); 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 (U_l)_l of E having no finite subcover of E. Use the total boundedness of E to define inductively a sequence of balls (B_n) in E with
   • the radius of B_n is 2^-n;
   • there is no finite subcover (U_l∩B_n)_l of B_n;
   • B_n+1∩B_n is not empty.
2. Let x_n be the center point of B_n and let y_n be any point in B_n+1∩B_n; hence we have d(x_n+1,x_n) <= d(x_n+1,y_n)+d(y_n,x_n) <= 2^-n-1+2^-n <= 2^-n+1. It follows for n <= p < q: d(x_p,x_q) <= d(x_p,x_p+1) + ... + d(x_q-1,x_q) <= 2^-p+1 + ... + 2^-q+2 <= 2^-n+2. Therefore, (x_n) 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 U_l0 contains a; since (x_n) converges to a and U_l0 is open, there is a large n such that the ball B_n is a subset of U_l0 - in contradiction to the construction of B_n.

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