# Proof of convergence by proving a sequence is Cauchy

• KPutsch
In summary, a sequence of real numbers, called Cauchy, is convergent if and only if it is Cauchy. A sequence is convergent if and only if it is not Cauchy and for every positive real number there is a positive integer such that for all natural numbers m, n > N |x_m - x_n| < epsilon. The triangle inequality states that |x + y| <= |x| + |y|.
KPutsch

## Homework Statement

Let 0 < r < 1. Let {A_n} be a sequence of real numbers such that |A_n+1 - A_n| < r^n for all naturals n. Prove {A_n} converges.

## Homework Equations

A sequence of real numbers is called Cauchy, if for every positive real number epsilon, there is a positive integer N such that for all natural numbers m, n > N |x_m - x_n| < epsilon.

A sequence is convergent if and only if it is Cauchy.

Triangle Inequality: |x + y| <= |x| + |y|

## The Attempt at a Solution

WLOG m>n, then |Am - An| = |Am - Am-1 + Am-1 - Am-2 ... -An|. Then we can use Triangle inequality to see that |Am - Am-1| < r^n, |Am-1 - Am-2| < r^n, ... . Then we can say that |Am - Am-1| + |Am-1 - Am-2| +...+ |An-1 -An|< n*r^n. Since r^n converges to zero, nr^n converges to zero, and nr^n < epsilon. Now to solve for n...

I have no idea how to solve nr^n < epsilon for n. Also, am I doing this right? I only have one poor example to go off of. If I can find n, will that mean I've shown that A_n is Cauchy?

Thank you for any help.

Depends on the professor, but mine just used:
Let E>0 be given. Choose N so that for n>N, nr^n<E.
Didn't solve for N at all, since we already know that nr^n converges.

You do not know n*r^n goes to zero. If you did you would be done.

|Am - Am-1| + |Am-1 - Am-2| +...+ |An-1 -An|< n*r^n

This is not true.

Also, since r is between 0 and 1 you know there is a natural number, say, a, such that 1/a <= r.

You know this because 1/n -> 0.

Ah, I proved in an earlier example that if 0 < r < 1, then nr^n converges to zero.

I realize now that the last pair of terms should be |An+1 -An|. The inequality should be true now: |Am - Am-1| + |Am-1 - Am-2| +...+ |An+1 -An|< n*r^n. So, am I done since I've shown that |An+1 - An| < ... < nr^n < epsilon? I really don't understand how this Cauchy sequence business works.

Last edited:
No, it's not.

$$|A_m-A_n|\le |A_m-A_{m-1}|+\dots+|A_{n+1}-A_n|<r^{m-1}+r^{m-2}+\dots+r^{n-1}<(m-n)r^{n-1}$$

Where the last inequality follows from the fact that r^m<r^n for m>n and noting there are m-n terms summed.

Read some more Cauchy proofs. Pay particular attention to how m and n are handled. Remember that there exists some natural number eta s.t 1/eta <= r and for any pi > eta, 1/pi < 1/eta. Also, n is not a free variable in the sense that it is bounded by m.

## 1. What is a sequence in mathematics?

A sequence in mathematics is a list of numbers that follow a specific pattern or rule. It can be written in the form of {an} where n represents the position of the number in the sequence and an represents the value of that number.

## 2. What is a Cauchy sequence?

A Cauchy sequence is a sequence in which the terms get closer and closer together as the sequence progresses. This means that for any given small number, there exists a point in the sequence where all subsequent terms are within that small distance from each other.

## 3. How do I prove that a sequence is Cauchy?

To prove that a sequence is Cauchy, you need to show that for any given small number, there exists a point in the sequence where all subsequent terms are within that small distance from each other. This can be done by using the definition of a Cauchy sequence and showing that it holds true for the given sequence.

## 4. What is the importance of proving convergence by proving a sequence is Cauchy?

Proving convergence by proving a sequence is Cauchy is important because it is a fundamental concept in analysis and calculus. It is used to show that a sequence of numbers will eventually approach a specific limit, which is important in many applications of mathematics.

## 5. Can a sequence be both Cauchy and divergent?

No, a sequence cannot be both Cauchy and divergent. If a sequence is Cauchy, it means that the terms are getting closer and closer together and therefore has a limit. If a sequence is divergent, it means that the terms are not approaching a specific limit and therefore cannot be Cauchy.

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