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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- Oct 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. - Oct 5th 2009, 01:04 PMPlato
- Oct 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 $\displaystyle (p_{n})$ is increasing and bounded above.

Consider the set of all points in the sequece $\displaystyle \{p_{n}|n\in\mathbb{N}\}$.

By the axiom of completeness, $\displaystyle sup\{p_{n}|n\in\mathbb{N}\}$ exists (since the set is a bounded subset of $\displaystyle \mathbb{R}$.

Assume $\displaystyle lim(p_{n})=s$, where $\displaystyle s=sup\{p_{n}|n\in\mathbb{N}\}$.

Then, for $\displaystyle \epsilon>0$, $\displaystyle s-\epsilon<s$. Thus, there exists a point $\displaystyle p_{N}$ such that $\displaystyle s-\epsilon<p_{N}<s$. Furthermore, because our sequence is increasing, it follows that, for every $\displaystyle n\geq\\N$, $\displaystyle p_{N}\leq\\p_{n}$. Thus, $\displaystyle s-\epsilon<p_{N}\leq\\p_{n}<s<s+\epsilon$.

This implies $\displaystyle |p_{n}-s|<\epsilon$ - Oct 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?

- Oct 6th 2009, 02:55 PMPlato
- Oct 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.