Bolzano-Weierstrass principle of choice extended towards ordinals

The Bolzano-Weierstrass principle of choice is the oldest method of the set theory, traditionally used in mathematical analysis. We are extending it towards transfinite sequences of steps indexed by ordinals. We are introducing the notions: hiker's tracks, hiker's maps and statements $P_n(X, Y, m)$; which are used similarly in finite, countable and uncountable cases. New proofs of Ramsey's theorem and Erd\"{o}s-Rado theorem are presented as some applications.