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Mathematics, 04.03.2020 23:26 jadentdaniels

Let (an) be a bounded sequence. Let S = {s ∈ R : ∃ a subsequence (ank ) converging to s}. This is called the set of subsequential limits. Bolzano-Weierstrass theorem implies there is at least one convergent subsequence, so S 6= ∅. Show S is bounded and lim sup an = sup(S).

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Let (an) be a bounded sequence. Let S = {s ∈ R : ∃ a subsequence (ank ) converging to s}. This is ca...
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