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 x1,x2, 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].

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