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Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence

Let H be the non-empty1 set of all accumulation points of a bounded real sequence (a_n)_{n\in\mathbb N}. The limit superior of a_n equals the supremum of H, formally \overline{\lim_{n\to\infty} a_n}=\sup H.

Proofs: 1


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References

Bibliography

  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983

Footnotes


  1. Please note that H is not empty due to the theorem of Bolzano-Weierstrass, since the sequence is bounded by hypothesis.