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I'm searching for a sequence with the following property:

$$\inf a_n < \lim \inf a_n < \lim \sup a_n < \sup a_n$$

I am looking for just one example, but I am not sure how to find one.

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  • $\begingroup$ Just take the first two terms so that they are inf and sup, for the remaining ones you take a simple alternating sequence squeezed between them. $\endgroup$
    – Karl
    Nov 27, 2018 at 15:14

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Here is an easy example : \begin{equation}\begin{aligned} a_1 &:= 1 \\ a_2 &:= 4 \\ a_{2n+1} &:= 2,~~~~~\text{for all $n \geq 1$} \\ a_{2n} &:= 3,~~~~~\text{for all $n \geq 1$}\end{aligned}\end{equation}

You can easily verify that \begin{equation}\begin{aligned} \inf a_n &= 1 \\ \liminf a_n &=2 \\ \limsup a_n &= 3 \\ \sup a_n &=4.\end{aligned}\end{equation}

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  • $\begingroup$ Ok thx, this is very helpful $\endgroup$ Nov 27, 2018 at 16:14

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