# Liminf and limsup of subsequences

Consider a bounded sequence $\{A_n\}_n$ and a subsequence $\{A_{n_k}\}_k \subseteq \{A_n\}_n$. Is it true that $$\liminf_{n\rightarrow \infty}A_n \geq \liminf_{k\rightarrow \infty}A_{n_k}$$ and $$\limsup_{n\rightarrow \infty}A_n \geq \limsup_{k\rightarrow \infty}A_{n_k}$$

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• The second one is ok, the first one should be with "$\le$" instead. You can derive it from the second one by using the identity $\liminf_n a_n=-\limsup_n(-a_n)$, for instance. – user228113 Dec 15 '15 at 12:45
• In general $\lim \sup A_n ≥ \lim \sup A_{n_k}$, but with $\lim \inf$ you have $≤$ instead of $≥$. – s.harp Dec 15 '15 at 12:45
• Check your inequalities, since $A_{n_k}$ is a subset of $A_n$, then $A_n$, with more elements, should have a smaller lower bound. – Michael Burr Dec 15 '15 at 12:46