Existence of subsequences $a_{n_k}$ that converges to a value between $\liminf a_n$ and $\limsup a_n$
No, take $a_n=(-1)^n$.
Then, $\liminf a_n =-1$, $\limsup a_n =1$.
However, any converging subsequence converges to either $-1$ or $1$.
No, take $a_n=(-1)^n$.
Then, $\liminf a_n =-1$, $\limsup a_n =1$.
However, any converging subsequence converges to either $-1$ or $1$.