# Show a sequence is Cauchy

1. Jan 30, 2012

### SpringPhysics

1. The problem statement, all variables and given/known data
If a sequence {xn} in ℝn satisfies that sum || xn - xn+1 || for n ≥ 1 is less than infinity, then show that the sequence is Cauchy.

2. Relevant equations
The triangle inequality?

3. The attempt at a solution
|| xm - xn || ≤ || Ʃ (xi+1 - xi) from i=n to m-1||
Using the triangle inequality and the given condition, I only get that the norm is less than infinity. I do not know how to transform this into an ε argument. Is there a property of finite sums of telescoping norms that would help?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jan 30, 2012

### genericusrnme

every convergent sequence in a metric space is a cauchy sequence
so you could either take that as a theorem or prove it (it's pretty much a one liner)
so you just want to show that your series converges

3. Jan 31, 2012

### SpringPhysics

I don't see how to show that the series converges if I am only given that the sum is finite. All I get from subtracting partial sums is that the norm of the difference is yet again finite...

4. Jan 31, 2012

### JHamm

If the sum of the distances between adjacent points converges then the distances must form a monotonically decreasing sequence such that $\lim \limits_{n\to\infty}||x_n - x_{n+1}|| = 0$ since distances are always non-negative.

What is the definition of a Cauchy sequence?

5. Jan 31, 2012

### SpringPhysics

So by definition of a limit,
lim n→∞ xn = xn+1
which means that the sequence converges to some limit L.

From here:
- I can use either that any convergent sequence in ℝn must be Cauchy
- or that the above implies that there is some N, M (natural numbers) such that

|| xn - L || < ε/2 for all n > N
|| xm - L || < ε/2 for all m > M

so that the definition of a Cauchy sequence is satisfied for all n, m > max{N, M}.

Is that correct?

By the way.....where do you find the code to format the limit? I couldn't find the latex for it.

6. Jan 31, 2012

### genericusrnme

A cauchy sequence is a sequence ${p_n}$ such that for every $\epsilon > 0$ there exists an integer $N$ such that $n,m>N$ implies $d(p_n,p_m) < \epsilon$

This means that the difference between each neighbouring members of the squence get smaller and smaller.

In the reals, suppose ${p_n} \rightarrow P$ then there exists N such that $n,m>N$
implies $d(P,p_n) < \frac{\epsilon}{2}$ and $d(P,p_m) < \frac{\epsilon}{2}$
Therefore $d(p_n,p_m) \leq d(P,p_n) + d(P,p_m) < \epsilon$
So every congergent sequence in R is a cauchy sequence

Can you see now what a cauchy sequence is?

7. Jan 31, 2012

### SpringPhysics

So that's basically what I said in my previous post, right?