Sequence $a_n$ such that $\inf a_n < \lim \inf a_n < \lim \sup a_n < \sup a_n$

Question

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.

Answer 1

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}