proof of convergence of a sequence with finite upcrossings

We show that a sequenceMathworldPlanetmath 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.

Denoting the infimum limit and supremum limitMathworldPlanetmath by

l=lim infnxn,u=lim supnxn,

then lu and the sequence converges to a limit if and only if l=u.

We first show that if the sequence converges then U[a,b] is finite for a<b. If l>a then there is an N such that xn>a for all nN. So, all upcrossings of [a,b] must start before time N, and we may conclude that U[a,b]N is finite. On the other hand, if la then u=l<b and we can infer that xn<b for all nN and some N. Again, this gives U[a,b]N.

Conversely, suppose that the sequence does not converge, so that u>l. Then choose a<b in the interval (l,u). For any integer n, there is then an m>n such that xm>b and an m>n with xm<a. This allows us to define infinite sequences sk,tk by t0=0 and


for k1. Clearly, s1<t1<s2< and xsk<a<b<xtk for all k1, so U[a,b]=.

Title proof of convergence of a sequence with finite upcrossings
Canonical name ProofOfConvergenceOfASequenceWithFiniteUpcrossings
Date of creation 2013-03-22 18:49:39
Last modified on 2013-03-22 18:49:39
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Proof
Classification msc 40A05
Classification msc 60G17