Proving Cauchy Sequence with Triangle Inequality

[SpringPhysics]

Homework Statement

If a sequence {x_n} in ℝⁿ satisfies that sum || x_n - x_n+1 || for n ≥ 1 is less than infinity, then show that the sequence is Cauchy.

Homework Equations

The triangle inequality?

The Attempt at a Solution

|| x_m - x_n || ≤ || Ʃ (x_i+1 - x_i) 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?

 

[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

 

[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...

 

[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?

 

[SpringPhysics]

So by definition of a limit,
lim n→∞ x_n = x_n+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

|| x_n - L || < ε/2 for all n > N
|| x_m - 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.

 

[genericusrnme]

So by definition of a limit,
lim n→∞ x_n = x_n+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

|| x_n - L || < ε/2 for all n > N
|| x_m - 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.

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?

 

[SpringPhysics]

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