For two positive vectors $a,b$ such that $a\prec b$, we know that there is an $m$ sequence of vectors $c^{(i)}$ such that $$a\prec c^{(1)}\prec \ldots \prec c^{(m)}\prec b$$ where each vector in the precedent formula differs from its successor by two entries. This is a classic result. I hope i don't need to explicit the algorithm. See {A. W. Marshall,I. Olkin, B. C. Arnold, Inequalities: Theory of Majorization and Its Applications}.

Usually the process begins from $b$ to arrive at $a$, i guess i found a way to start from $a$ and arrive at $b$ and by some «example verification» i obtained the same sequence $c^{(i)}$. For example in the literature they first obtain $c^{(m)}$ and i obtain first $c^{(1)}$. Before expliciting the small process can anyone justify or obtain a similar algorithm, how to get first $c^{(1)}$ then $c^{(2)}$ and finally $b$.

I'll post the route if everything is clear. The question is for justification not more.