First proved by Dirichlet, its holds for real numbers, complex numbers, and p-adic numbers, but fails for the rational numbers and for infinite dimensional normed vector spaces.
First proved by Dirichlet, its holds for real numbers, complex numbers, and p-adic numbers, but fails for the rational numbers and for infinite dimensional normed vector spaces. This theorem's namesake formulated it in 1895 parallel to the concept of uniform continuity. Holding that if a closed set of points on a line can be covered by a set of intervals so that every point of the set is an interior point of at least one of the intervals, then there exist a finite number of intervals with the covering property, it is easily derived via the Cantor's intersection and Bolzano-Weierstrass theorems, FTP name this theorem from analysis which equates the compactness of set to its being closed and bounded.
First proved by Dirichlet, its holds for real numbers, complex numbers, and p-adic numbers, but fails for the rational numbers and for infinite dimensional normed vector spaces.
{Heine-Borel} Theorem