Baroni’s theorem

Let ${(x_n)}_{n\ge 0}$ be a sequence of real numbers such that $\underset{n\to \mathrm{\infty }}{lim}(x_{n+1}-x_n)=0$. Let $A=\{x_n|n\in \mathbb{N}\}$ and A’ the set of limit points of $A$.Then A’ is a (possibly degenerate) interval from $\overline{ℝ}$, where $\overline{ℝ}=ℝ\bigcup \{-\mathrm{\infty },+\mathrm{\infty }\}$