This question revisits 2.4.7 from Abbott. Remind yourself of the definition of lim sup. Let (an)...

This question revisits 2.4.7 from Abbott. Remind yourself of the definition of lim sup. Let (a_n) be a bounded sequence. Let S = {s ∈ R : ∃ a subsequence (a_nk ) converging to s}. This is called the set of subsequential limits. Bolzano-Weierstrass theorem implies there is at least one convergent subsequence, so S cannot equal ∅. Show S is bounded and lim sup a_n=sup(S).

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Colby Messinger answered 6 days ago

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