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.

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 $D[a,b]$.

Title convergence of a sequence with finite upcrossings ConvergenceOfASequenceWithFiniteUpcrossings 2013-03-22 18:49:36 2013-03-22 18:49:36 gel (22282) gel (22282) 4 gel (22282) Theorem msc 40A05 msc 60G17 UpcrossingsAndDowncrossings