= Doubly stochastic realization of summable-sequence majorization
{title2=$x_i=\sum_jp_{ij}y_j$}
Decreasing strictly positive <summable sequences> with equal totals and $x\prec y$ admit an infinite <doubly stochastic matrix> realizing the displayed relation. Choose $k$ with $y_k\geq x_1>y_{k+1}$ and average the pair to obtain $x_1$ and $y_k+y_{k+1}-x_1$. The remaining decreasing sequence majorizes the remaining target sequence: short initial sums contain values at least $x_1$, and longer sums are the old sums minus $x_1$. Repeat on the tail, freezing one normalized finite-support row at each stage. Limit column sums are at most one; <Tonelli's theorem>, equal finite totals and strict positivity of every $y_j$ force every column sum to equal one.
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