PlanetMath: Baroni's theorem

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Baroni's theorem

(Theorem)

Let $(x_n)_{n\geq 0}$ be a sequence of real numbers such that $\displaystyle \lim_{n \rightarrow \infty} (x_{n+1}-x_n) = 0$ . Let $A=\{ x_n \vert n \in \mathbb{N} \}$ and A' the set of limit points of . Then A' is a (possibly degenerate) interval from $\overline{\mathbb{R}}$ , where $\overline{\mathbb{R}}=\mathbb{R} \bigcup \{-\infty,+\infty\}$

"Baroni's theorem" is owned by mathwizard. [ full author list (2) | owner history (1) ]

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Attachments:: Proof of Baroni's theorem (Proof) by mathwizard

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Cross-references: interval, limit points, real numbers, sequence

This is version 3 of Baroni's theorem, born on 2003-04-03, modified 2004-05-24.
Object id is 4141, canonical name is BaronisTheorem.
Accessed 1454 times total.

Classification:

AMS MSC:

40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)

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