Limited Omniscience and the Bolzano-weierstrass Principle

The constructive study of metric spaces requires at first an examination of each classical proposition for numerical content. In classical mathematics it is a theorem that sequences in a compact space have convergent subsequences, but this is not constructively true. For compact intervals on the real line it has long been known that this theorem is nonconstructive because it implies the Limited Principle of Omniscience (LPO); here we show that it is equivalent to LPO. At the same time, we obtain other equivalent forms of LPO which concern arbitrary sequences of positive integers. We follow the strict constructive approach of Errett Bishop [1], discussions of which are found in [4] and [6]. The lack of numerical content in a classical proposition is shown by relating it to a nonconstructive omniscience principle; the result is called a Brouwerian counterexample. Brouwerian counterexamples are discussed in [1], [2], Section 2 of [3] and in [5].