Show something is a Cauchy sequence

Question

If I want to show that something is Cauchy, should I show it converges and then show it is Cauchy or should I go at it straight from the definition. I am just trying to figure out generally what to do for a sequence.

I don't have a concrete example. I am sorry, but like if the question was Prove that this series is Cauchy... How would I go about doing it?

Different somethings will require different techniques. Give a concrete example of what you have in mind. — lhf, May 13 '14 at 01:27

score 2 · Accepted Answer · answered May 13 '14 at 01:32

As a general rule:

If you know what it converges to, i.e. if what it converges to is obvious, it is often easier to check convergence directly. For instance, the sequence $1 - \frac{1}{n}$ obviously converges to $1$, and it's easier to check this than it is to check the Cauchy criterion.
If you have no idea what it converges to, the Cauchy criterion is a valuable tool. This is for example the most elementary way to show the partial sums of a series converge.

score 1 · Answer 2 · answered May 13 '14 at 01:30

1

If a sequence does converge, then it is a Cauchy sequence. Be careful, though, that not always does a Cauchy sequence converges. For exemple, the sequence $x_n = \frac{1}{n}$ is a Cauchy sequence, but does not converge in $]0,1]$. It does converge in $[0,1]$. Search about completeness of a topological space (or a metric space, for what it's worth).

answered May 13 '14 at 01:30

Ivo Terek

77,665

if its a Cauchy sequence it converges. thats a theorem – Jack Armstrong May 13 '14 at 01:32
In $\mathbb{R}$ with the usual topology yes, but not in all metric spaces. – Thomas Credeur May 13 '14 at 01:33
oh okay, I am only dealing with the real numbers and we have not defined metric spaces so I understand. – Jack Armstrong May 13 '14 at 01:35
Forget the rest of $\mathbb{R}$ exists, look only at $]0,1]$. The sequence does not converge because $0 \notin ]0,1]$. – Ivo Terek May 13 '14 at 01:37

score 0 · Answer 3 · answered May 13 '14 at 01:29

If you have a candidate for what the sequence converges to, then it can be easier to show that the sequence converges. If you don't, though, it can be hard to show that a sequence converges just from the definition. In that case it can be easier to use the definition of Cauchy

Show something is a Cauchy sequence

3 Answers3