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There is a lemma for a certain proof in my notes:

If $\sum_n |a_n| < \infty$ and $\sum_n |b_n| < \infty$ then $\sum_{n=0}^\infty \sum_{k=0}^n|a_k||b_{n-k}| < \infty$

Denote $ A_N = \sum_{n=0}^N |a_n|$ and $B_N = \sum_{n=0}^N|b_n|$ then since $A_N$ and $B_N$ is bounded and non negative then $A_N \to A$ and $B_N \to B$ as $N\to\infty$

$\sum_{n=0}^N\sum_{k=0}^n|a_n||b_{n-k}| \leq \sum_{n=0}^N\sum_{m=0}^N|a_n||b_m|$

I don't understand the step there, how is the LHS less than or equal to the right hand side?

To be honest, I'm overall fairly confused with the cauchy product, I keep seeing $\sum_n a_nb_{n-k}$ everywhere, is it equal to $\sum_na_n \sum_m b_m$ or something similar? Really not sure...

thanks

DHx
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  • Have you taken two convergent series of positive terms and multiplied them together? I mean by hand, just the first few terms? If not, I suggest that you try it and see what’s going on. There’s nothing mysterious about the “Cauchy product”, when you’ve done a little computation, it looks like the most natural thing in the world. – Lubin Dec 30 '13 at 04:33

1 Answers1

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The reason the inequality is true is because there are more elements summed over on the right than the left and they are all non-negative. Also $\sum_{n=0}^{\infty}\sum_{m=0}^{n}a_{m}b_{n-m}\neq\big(\sum_{n=0}a_{n}\big)\big(\sum_{m=0}^{\infty}b_{m}\big)$. Notice that the sums on the right sums $n$ and $m$ independently while on the left the sum over $m$ depends on how large $n$ is. There could also be issues of sign of the numbers involved in the sums.

user71352
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