Not sure how to go about this one.

If pn is a non-decreasing sequence and there is a point, p, to the right of each point of the sequence, then the sequence converges to some point.

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- October 5th 2009, 12:54 PMspikedpunchSequence convergence
Not sure how to go about this one.

If pn is a non-decreasing sequence and there is a point, p, to the right of each point of the sequence, then the sequence converges to some point. - October 5th 2009, 01:04 PMPlato
- October 5th 2009, 02:06 PMDanneedshelp
Isn't this the Monotone convergence theorem?

"If a sequence is monotone and bouned, then is converges"

here is an outline:

Suppose the sequence is increasing and bounded above.

Consider the set of all points in the sequece .

By the axiom of completeness, exists (since the set is a bounded subset of .

Assume , where .

Then, for , . Thus, there exists a point such that . Furthermore, because our sequence is increasing, it follows that, for every , . Thus, .

This implies - October 6th 2009, 02:00 PMspikedpunch
If I'm trying to prove that pn converges to some point, then I don't think I can assume that lim(pn)=s ......right?

- October 6th 2009, 02:55 PMPlato
- October 6th 2009, 07:46 PMspikedpunch
I'm saying that if I'm trying to prove a sequence converges, i can't assume that it has a limit and converges, because I would be assuming what I'm trying to prove.