# Proving Increasing Sequence or Showing Convergence of {a_n}

• mynameisfunk
In summary, the problem discusses a bounded sequence of real numbers and shows that if the sequence satisfies the inequality a_n \leq \frac{a_{n-1}+a_{n+1}}{2}, then the sequence b_n = a_{n+1}-a_n is increasing. It also asks to show that if the inequality does not hold, then the sequence a_n converges. However, it is not possible to determine if a_n converges or not based on the given information.
mynameisfunk

## Homework Statement

Suppose that {$a_n$} is a bounded sequence of real numbers such that, for all $n$, $a_n \leq \frac{a_{n-1}+a_{n+1}}{2}$. Show that $b_n=a_{n+1}-a_n$ is an increasing sequence. Otherwise show that {$a_n$} converges.

## The Attempt at a Solution

I do not see it at all..
It seems that since $2a_n \leq a_{n-1}+a_{n+1}$ that the sequence could be either increasing OR decreasing. i could fit the set of integers, either increasing or decreasing and the inequality holds and $b_n$ is stagnate. BUT, since {$a_n$} is bounded, it must be convergent?? i couldn't find any theorems or anything to support this claim though. Possibly "For a real valued sequence {$s_n$}, $\lim_{n \rightarrow \infty}=s$ iff it's lim sup=lim inf =s (as n approaches infinity)" ??

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mynameisfunk said:

## The Attempt at a Solution

BUT, since {$a_n$} is bounded, it must be convergent?? i couldn't find any theorems or anything to support this claim though.

Well your claim is not quite correct; that is why you couldn't find a theorem to support it.

There are tons of bounded sequences that are not convergent.

Eg
(-1,1,-1,1...)

I have not tried the problem but I think I see what you mean by the sequence can be either decreasing or increasing.

I see 3 things
1) If the sequence is increasing then it is increasing at an increasing rate and is therefore not bounded.
2) If it is decreasing it is decreasing at a decreasing rate and could possible converge.
3) If it is constant then we know it converges.

I believe your best bet is to show that the sequence is cauchy. For one thing we know the sequence is not increasing. :)

is it possible to take $n$ to $-\infty$? We have never done that before..

mynameisfunk said:
is it possible to take $n$ to $-\infty$? We have never done that before..

Hey mate.
I made a mistake; b_n is actually increasing by definition. b_n is the distance between sucessive points and if you rearrange the first inequaility it is extremely easy to see that b_n-1 <= b_n which is what we want.

I actually arrived at that result in my original post but I was too stupid to notice that (1) was, in fact, the desired result.

Sometimes extremely obvious things are hard to see :-) :-).

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Ya, I was able to arrive at that as well, but I am kind of confused by what I am supposed to do in the 2nd part. $a_n$ shouldn't converge since $a_n-a_{n-1} \leq a_{n+1}-a_n$, right?? But since $a_n$ is bounded, I suppose the $b_n$ must be bounded above?

The question ask you to either shown b_n is increasing or a_n is convergent. You have showed that b_n is increasing so you are done.

You do not have enough info to say that a_n converges. In fact I don't think it does.

OK, well what about this. Can i deduce from the fact that $a_n-a_{n-1} \leq a_{n+1}-a_n$ and that $a_n$ is bounded and is either monotonically increasing or decreasing and that every monotonically increasing/decreasing sequence converges iff it is bounded?

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I don't think so but you should probably get a second opinion.

{a_n} can't converge. It's not Cauchy : ).

EDIT: Well, I guess if they were all negative terms, and b_n was increasing to zero, then maybe.

l'Hôpital said:
{a_n} can't converge. It's not Cauchy : ).

EDIT: Well, I guess if they were all negative terms, and b_n was increasing to zero, then maybe.

b_n increases to zero is the key. That gives a_n convergent.

## 1. What is an increasing sequence?

An increasing sequence is a sequence of numbers where each term is larger than the one before it. This means that as you move through the sequence, each number is getting bigger and bigger.

## 2. How do you prove that a sequence is increasing?

To prove that a sequence is increasing, you can show that each term in the sequence is greater than or equal to the previous term. This can be done using mathematical induction or by directly comparing each term in the sequence.

## 3. What is convergence of a sequence?

Convergence of a sequence refers to the idea that as you move through the sequence, the terms are getting closer and closer to a certain value, known as the limit. In other words, the sequence is approaching a specific value as the number of terms increases.

## 4. How do you show convergence of a sequence?

To show convergence of a sequence, you can use various methods such as the squeeze theorem, the ratio test, or the comparison test. These methods involve comparing the sequence to other known sequences and using mathematical operations to determine the limit of the sequence.

## 5. What is the importance of proving the convergence of a sequence?

Proving the convergence of a sequence is important because it allows us to determine the behavior of the sequence as the number of terms increases. This information can be used to make predictions and solve problems related to the sequence.

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