Find a sequence such that $a_{2n} \leq a_{2n+2} \leq a_{2n+3} \leq a_{2n+1}$ for all $n \geq 0$ which does not converge

Question

How to construct such a sequence? please help. thank you!

Answer 1

If you want a sequence with strict inequalities, you can go for a variation on @Dark's comment above, something like $$ (-1)^{n+1}\left(1+\frac1{n+1}\right) $$

Answer 2

Idea: if $n$ is even, choose a nondecreasing sequence (for example, constant). If $n$ is odd, choose a descreasing sequence that is bounded from below by the other one (and has a different limit). For instance, $a_n=1$ if $n$ is even, $a_n=2+\frac{1}{n}$ if $n$ is odd.