Mandelkern shows that this theorem is nonconstructive due to its equivalence to the Limited Principle of Omniscience, which is duplicated by its Brouwerian counterexamples. Its namesake property for ideals is equivalent to a P-point of the Cech-Stone compactification beta-omega, but it fails for ideals when their quotient Boolean algebra has a countably splitting family. Its extension to transfinite step sequences with ordinal indices leads to a proof for the Erdos-Rado theorem. (*) One of its common proofs generates a series of nested intervals, each of which contains the leftmost subinterval of the last, and demonstrates that their left end points are a Cauchy sequence with a limit about which the subinterval contains infinitely many points of an infinite set. It follows from subsequence
Mandelkern shows that this theorem is nonconstructive due to its equivalence to the Limited Principle of Omniscience, which is duplicated by its Brouwerian counterexamples. Its namesake property for ideals is equivalent to a P-point of the Cech-Stone compactification beta-omega, but it fails for ideals when their quotient Boolean algebra has a countably splitting family. Its extension to transfinite step sequences with ordinal indices leads to a proof for the Erdos-Rado theorem. (*) One of its common proofs generates a series of nested intervals, each of which contains the leftmost subinterval of the last, and demonstrates that their left end points are a Cauchy sequence with a limit about which the subinterval contains infinitely many points of an infinite set. It follows from subsequences of real numbers via the Monotone convergence axiom, and may be derived from Cantor's intersection theorem and the Heine-Borel theorem, to which it is equivalent. FTP, name this algebraic theorem stating every bounded infinite sequence contains a convergent subsequence, named for a Czech and a German mathematician.
Mandelkern shows that this theorem is nonconstructive due to its equivalence to the Limited Principle of Omniscience, which is duplicated by its Brouwerian counterexamples.
{Bolzano}-{Weierstrass Theorem}[{accept}: Heine-Borel Theorem until it is mentioned]