I have the following theorem:
Suppose $a_n$ is a bounded sequence and $u=\limsup(a_n)$. Then:
i) There is a subsequence $a_{n_k}$ with $a_{n_k}\rightarrow u$
ii) If $a_{m_k}$ is convergent subsequence with limit x, then $x\leq u$
Do we have an analogous result for $\liminf (a_n)$? (I'm pretty sure we do but it's not in my notes which seems strange)
Thanks very much for any help