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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If the sub-sequence has an accumulation point (infinitely many elements are arbitrarily close to it) then it is an accumulation of the original sequence. So, the lowest accumulation point of the sequence is as low as the lowest accumulation of the sub-sequence and can be even lower. Similarly, the highest accumulation point of the sequence is as high as the highest accumulation point of the sub-sequence and can be even higher.