Using a comprehensive approach, this paper aims to demonstrate, clearly and rigorously, the validity of the Collatz conjecture. To this end, the original 3n+1 iteration is reformulated by isolating the odd terms into sequences referred to as R-Cz sequences. These sequences are analyzed through their structural properties and their distribution among the odd natural numbers. As a first essential result, it is shown that they do not admit non-trivial cycles: the only possible cycle is the trivial one, of value and length 1. Two independent proofs that all R-Cz sequences converge are then presented. The first, combinatorial in nature, relies on the finiteness of intervals that could possibly separate terms of the sequences. The second, set-theoretic, is based on a contradiction between the countability of the odd integers and the uncountable cardinality of the hypothetical divergent R-Cz sequences. Both methods lead to the same conclusion: all Collatz sequences eventually enter the cycle (1,4,2).