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Suppose that $a_n$ and $b_n$ converge to $\alpha$ and $\beta$ as $n\to\infty$ respectively. Show that: $$\frac{a_0b_n+a_1b_{n-1}+...+a_nb_0}{n}$$

converges to $\alpha\beta$ as $n\to\infty$

Resolution: Let $M$ be an upper bound of the two convergent sequences $|a_n|$ and $|b-n|$. For any $\epsilon>0$ we can take a positive integer $N$ satisfying $|a_n-\alpha|<\epsilon$ and $|b_n-\beta|<\epsilon$ for all integers $n>N$. If $n>N^2$, then:

$|a_k-b_{n-k}-\alpha\beta|\leqslant|(a_k-\alpha)b_{n-k}+\alpha(b_{n-k}-\beta)|\leqslant(M+|\alpha|)\epsilon$

for any integer $k\in[\sqrt{n},n-\sqrt{n}]$. Therefore: $$|\frac{1}{n}\sum_\limits{k=0}^{n}a_kb_{n-k}-\alpha\beta|\leqslant\frac{1}{n}\sum_\limits{\sqrt{n}\leqslant k\leqslant n-\sqrt{n}}^{n}a_kb_{n-k}-\alpha\beta\\+2(\alpha\beta+M^2)\frac{\sqrt{n}+1}{n}\leqslant(M+|\alpha|)\epsilon+2(\alpha\beta+M^2)$$

we can take $n$ so large that the last expression is less than $(M+|\alpha|+1\epsilon$

Questions:

1) I have just copied the resolution. However I think that here $\frac{1}{n}\sum_\limits{\sqrt{n}\leqslant k\leqslant n-\sqrt{n}}^{n}a_kb_{n-k}-\alpha\beta\\+2(\alpha\beta+M^2)\frac{\sqrt{n}+1}{n}$

there should not be a sum once the author is using$2(\alpha\beta+M^2)\frac{\sqrt{n}+1}{n}$. What do you think?

2) How did the author derive $\frac{\sqrt{n}+1}{n}$?

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The author divides the sum $\sum_{k=0}^n$ into two: one with $k\notin[\sqrt{n},n-\sqrt{n}]$ and one with $k\in[\sqrt{n},n-\sqrt{n}]$. The first one has at most $2\,(\sqrt{n}+1)$ terms (from $0$ to $\sqrt n$ and from $n-\sqrt n$ to $n$), and each term is bounded by $M^2+|\alpha\,\beta|$. Thus$$ \Bigl|\frac{1}{n}\sum_{k\notin[\sqrt{n},n-\sqrt{n}]}a_kb_{n-k}-\alpha\beta\,\Bigr|\le\frac{1}{n}\,2\,(\sqrt{n}+1)(M^2+|\alpha\,\beta|). $$ For the second each term is a most (M+|\alpha|)\,\epsilon and there are less than $n$ terms. Then $$ \Bigl|\frac{1}{n}\sum_{k\in[\sqrt{n},n-\sqrt{n}]}a_kb_{n-k}-\alpha\beta\,\Bigr|\le\frac{1}{n}\,\Bigl(\sum_{k\in[\sqrt{n},n-\sqrt{n}]}1\Bigr)(M+|\alpha|)\,\epsilon\le(M+|\alpha|)\,\epsilon. $$

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  • $\begingroup$ Thanks for your answer! How do you know that $k\notin[\sqrt{n},n-\sqrt{n}]$ has $2(\sqrt{n}+1)$ terms at the maximum? Why is the 1 summed to the later expression? Why is it not only $2\sqrt{n}$? $\endgroup$ – Pedro Gomes Sep 1 '18 at 10:36
  • $\begingroup$ Sorry for bothering again but on the expression $(\sum_{k\in[\sqrt{n},n-\sqrt{n}]}1\Bigr)(M+|\alpha|)\,\epsilon$ How does $(\sum_{k\in[\sqrt{n},n-\sqrt{n}]}1$ vanish? Thanks in advance! $\endgroup$ – Pedro Gomes Sep 1 '18 at 10:45
  • $\begingroup$ 1. $\sqrt n +1$ between 0 and $\sqrt n$ , and the same on the other extreme of the interval. 2. Remember that it is divided by $n$. $\endgroup$ – Julián Aguirre Sep 2 '18 at 13:40

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