Suppose that $\langle b_n\rangle$ and $\langle c_n\rangle$ be two sequences which are bounded and we know their limit superiors and limit inferiors.If we define a new sequence $\langle a_n\rangle$ as $\langle a_n\rangle=\langle b_nc_n\rangle$. My question is "Can we always conclude limit superior and limit inferior of $\langle a_n\rangle$?
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Martin Sleziak
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Styles
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No this is in general not possible, unless the limit of $b_n$ or $c_n$ exists. But, we can always get some upper bounds and lower bounds in terms of the liminf and limsups of the sequences. – clark May 21 '18 at 21:53
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1For positive sequences, you can at least tell som upper and lower bound: $\liminf b_n \cdot \limsup c_n \le \limsup (b_nc_n) \le \limsup b_n \cdot \limsup c_n$. You can find more about one of these inequalities here: lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $ – Martin Sleziak May 28 '18 at 12:06
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No. Suppose $b_{2n}=1, b_{2n+1}=0, c_{2n}=0, c_{2n+1}=-1 $. Then $b_nc_n = 0$ for all $n$.
marty cohen
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