The bijectivity criterion, continuum hypothesis, and number sequence and series without some dogmas

N Introduction, we give the Alternative decision of David Hilbert’s first Problem. Our paper contains demonstrative denying a hypothesis about the existence of a bijection between a set of positive integers and its own subset. This statement is a basis of an alternative methodology, in which a significant tool is the concept of С- ( m, k )- pair of natural variables. We define e-divergence and w-convergence of number sequences with this methodology. In particular, the equality is a characteristic feature for a w-converging number sequence. We proved that the set of Cauchy sequences coincides with the set of w-converging ones and, hence, contains a subset of the infinite large sequences; everyone from them converges to corresponding infinite large number ( ILN ). In particular, a harmonic series converges to the some ILN, and the necessary attribute of some number series convergence is also a sufficient one.