Heine–Borel theorem

The Heine-Borel Theorem in analysis states:

A subset of the real numbers R is compact if and only if it is closed and bounded.

This 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 theorem is closely related to the theorem of Bolzano-Weierstrass.

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.

Proof

Here, we consider the version for the real numbers, as stated above.

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If a subset of the real numbers is compact, then it is bounded.

Let X be a compact subset of the real numbers.

Define a collection of open sets O_n to be the open interval (-n, +n). These sets cover X. Some finite sub-collection must also cover X. This finite sub-collection is bounded, and so X is bounded.

If a subset of the real numbers is compact, then it is closed.

Suppose (for a contradiction) that it were not closed. Then, there would be a sequence of number x_n in X such that x_n converges to a point x not in the set X.

Define the sequence of open sets O_n where O_n = X - {x_n, x_n+1, x_n+2, ...).Now, O_n covers X, because x_n coverges to a point not in X. So, some finite sub-collection of O_n covers X_n.

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