convergence of a sequence with finite upcrossings

The following result characterizes convergence of a sequence in terms of finiteness of numbers of upcrossings.

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 $a<b$ .

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 $D[a,b]$ .

单词	ConvergenceOfASequenceWithFiniteUpcrossings
释义	convergence of a sequence with finite upcrossings The following result characterizes convergence of a sequence in terms of finiteness of numbers of upcrossings. 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 $a<b$ . 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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