What relation, if any, can you state for the lim sups and lim infs of a sequence {an} and one of its subsequences {$a_n$$_k$}?
2026-03-26 21:25:56.1774560356
What are the lim sups and lim infs of a sequence?
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This is probably what you're looking for:$$ \liminf_{n \to \infty} \{a_{n}\} \leq \liminf_{k \to \infty} \{a_{n_k}\} \leq \limsup_{k \to \infty} \{a_{n_k}\} \leq \limsup_{n \to \infty} \{a_{n}\} $$