  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.
