40A05-BaronisTheorem.tex

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\pmtitle{Baroni's theorem}
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\begin{document}
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 $A$.
Then A' is a (possibly degenerate) interval from $\overline{\mathbb{R}}$, where $\overline{\mathbb{R}}=\mathbb{R} \bigcup \{-\infty,+\infty\}$
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