# Sequence convergence

• October 5th 2009, 12:54 PM
spikedpunch
Sequence convergence

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 PM
Plato
Quote:

Originally Posted by spikedpunch
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.

That is an oddly stated problem.
Mathematically translated it says “A non-decreasing sequence bound above converges”.
Which of course is true. The proof follows from the completeness property.
The sequence converges to its least upper bound.
• October 5th 2009, 02:06 PM
Danneedshelp
Quote:

Originally Posted by spikedpunch

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.

Isn't this the Monotone convergence theorem?

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

here is an outline:

Suppose the sequence $(p_{n})$ is increasing and bounded above.

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

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

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

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

This implies $|p_{n}-s|<\epsilon$
• October 6th 2009, 02:00 PM
spikedpunch
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 PM
Plato
Quote:

Originally Posted by spikedpunch
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?

I really don't know how to answer that question.
Because I don't know what it means.
It is quite easy to prove this theorem: A bounded non-decreasing sequence converges to its least upper bound.
• October 6th 2009, 07:46 PM
spikedpunch
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.