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 $x_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{,}\mathrm{\dots }$ of real numbers converges to a limit in the extended real numbers if and only if the number of upcrossings $U\mathit{}\mathrm{[}a\mathrm{,}b\mathrm{]}$ is finite for all .

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
Canonical name	ConvergenceOfASequenceWithFiniteUpcrossings
Date of creation	2013-03-22 18:49:36
Last modified on	2013-03-22 18:49:36
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	4
Author	gel (22282)
Entry type	Theorem
Classification	msc 40A05
Classification	msc 60G17
Related topic	UpcrossingsAndDowncrossings