40A05-ConvergenceOfASequenceWithFiniteUpcrossings.tex

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\pmtitle{convergence of a sequence with finite upcrossings}
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The following result characterizes convergence of a sequence in terms of finiteness of numbers of upcrossings.

\begin{theorem*}
A sequence $x_1,x_2,\ldots$ of real numbers converges to a limit in the extended real numbers if and only if the number of upcrossings $U[a,b]$ is finite for all $a<b$. 
\end{theorem*}

Since the number of upcrossings $U[a,b]$ differs from the number of downcrossings $D[a,b]$ by at most one, the theorem can equivalently be stated in terms of the finiteness of $D[a,b]$.

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