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evinda

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Let $(a_n)$ be a bounded sequence such that $\inf_{\ell} a_{\ell}<a_n< \sup_{\ell} a_{\ell}$ for each $n=1,2, \dots$

I want to show that there are subsequences $(a_{k_n})$ increasing and $(a_{m_n})$ decreasing such that $a_{k_n} \to \sup_{\ell} a_{\ell}$ and $a_{m_n} \to \inf_{\ell} a_{\ell}$.Could you give me a hint how we could find the desired subsequences? (Thinking)