If $\sum{r_j}$ is an absolutely convergent series bounded by R, and the sequence $\{s_{j,k}\}$ is bounded by S and $\lim_{k \to \infty} s_{j,k} = 0$, then how do we show that $\lim_{n \to \infty} \sum^{\infty}_{j=0}r_j s_{j,k}$ go to $0$? By the answer here Under what condition we can interchange order of a limit and a summation? I can see that we can interchange the limit and the sum, but I was wondering what you need to be able to show this a bit more intuitively.
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The Dominated Convergence Theorem will justify this rigorously. If $|s_{j,k}|\le S$ always, then each series is dominated by $\sum S|r_j|$ which converges by hypothesis; therefore the limit can be taken inside the infinite series.
Greg Martin
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Yes, but how can it be shown using more elementary methods? It is my first year as a student in analysis, and by this point we have not gotten to integrals. – beedo Dec 09 '20 at 22:34
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The proof of the Dominated Convergence theorem uses more elementary methods, so if you really wanted to, you could pull the required ideas from the proof and do this example by hand. – Greg Martin Dec 09 '20 at 22:36