This question is about extending a result on transportation polytopes from Brualdi regarding $m\times n$ matrices to the case when $m=\infty$.

Notation: Denote an $m\times n$ matrix by $A=[a_{i,j}]$, and consider sets $K\subset\lbrace 1,2,\ldots, m\rbrace$ and $J\subset \lbrace 1,2,\ldots, n\rbrace$. Define $$A[K,J):=[a_{i,j}:i\in K, j\in\lbrace 1,2,\ldots,n\rbrace-J].$$

Denote by $N(R,S)$ by the class of all $m\times n$ nonnegative matrices with row sum given by the vector $R=(r_1,\ldots, r_m)$ and column sum given by the vector $S=(s_1,\ldots, s_n)\,\,\,$ ($N(R,S)$ is called a transportation polytope).

The following theorem is from Brualdi's Combinatorial Matrix Classes:

I am trying to construct an $\infty\times n$ matrix with $\sum_{m=1}^{\infty}r_m = \sum_{j=1}^ns_j=1.$ Will this theorem also hold when $m=\infty$? I tried extending the proof to the case $m=\infty$, but Brualdi's proof heavily relies on the fact that $m$ is finite. My idea was to do truncation on the number of rows and take a limit as $m\to \infty$, but I am unsure if that works.