Bounded divergence for every regular summability matrix

For every regular summability matrix $A$, there is a bounded sequence whose transformed row sums do not converge. A gliding hump argument selects successive rows whose mass on already assigned coordinates is small and whose remaining tails are small. On each new finite block, choose signs agreeing or disagreeing with that row alternately. Since row sums tend to one, the transformed values have subsequences uniformly above and below zero. This does not contradict preservation of ordinary convergence, since the constructed sequence is divergent.