Assume that $a_n$ and $b_n$ are 0-1 sequences such that $$ \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n = p. $$ Let also $c_n$ an other 0-1 sequence. Is it true that $$ \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n c_n = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n c_n . $$ I think it is true because if the first equation is correct, than the limit is also true on each subsequence and in particular in the one where $c_n=1$. Is my argument correct?
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Not quite a duplicate question, but the answers there also answer this question. – Daniel Fischer Apr 05 '18 at 12:53
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1I think it's worth noting that it's not true that if a sequence has some Cesaro limit, then so do its subsequences. That's a property unique to standard limits. – Wojowu Apr 05 '18 at 14:20
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@Wojowu yes indeed, that was my mistake. – user52227 Apr 05 '18 at 21:40
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No, the claimed property does not hold. Take $a_n=1$ iff $n$ is even and $b_n=1$ iff $n$ is odd. Then $$\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n = \frac{1}{2}.$$ Now if $c_n=a_n$ then $a_nc_n=a_n$ whereas $b_nc_n=0$ and it follows that $$\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n c_n =\frac{1}{2}\not=0 =\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n c_n .$$
Robert Z
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