Problem 11_ Let Sn be a bounded sequence in R. Let A be the set of a R such that {n € N : Sn < a} is finite, i.e . all but finitely many Sn are ≥ a. Let B be the set of b € R such that {n € N Sn > b} is finite: Prove sup A = lim inf_ Sn and inf B = lim sup Sn"
Added by Christian M.
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The limit inferior of $(s_n)$ is defined as: $$\liminf_{n \to \infty} s_n = \lim_{n \to \infty} \left(\inf_{k \geq n} s_k\right)$$ Similarly, the limit superior of $(s_n)$ is defined as: $$\limsup_{n \to \infty} s_n = \lim_{n \to \infty} \left(\sup_{k \geq n} Show more…
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