what is the relation between Cauchy sequence and convergent sequence?

Question

Can someone summarize all the cases of Cauchy sequence and convergent sequences?

I mean I know that:

1. Convergent sequence $\Rightarrow$ Cauchy sequence

2. Cauchy not necessarily $\Rightarrow$ Convergent sequence

what about divergent sequence?

is the following true?

3. Divergent sequence $\Rightarrow$ Not Cauchy sequence

4. Not Cauchy sequence not necessarily $\Rightarrow$ Divergent sequence

So what's true and what's not?

Answer 1

For the two cases you have listed here, the answer depends on in which space your working in.

In order to talk about a Cauchy sequence you need a metric. A space that is metric and in which all Cauchy sequences converge is called a complete metric space. Note that if a metric space is not complete it doesn't mean that a Cauchy sequence never converges, it only means that there is at least one case of a sequence that is a Cauchy sequence but does not converge. So, in a complete metric space convergence and Cauchy are equivalent attributes of a sequence, in a metric space that is not complete only the implication 1) is true.

This means for your listed cases:

3. is true in a complete metric space and not true in general in a not complete metric space. For in a not complete metric space there is at least one sequence that is divergent but also Cauchy (this ist he reason for it not being complete).

4. is true in every metric space. This follows directly from 1). For if there would be a not Cauchy sequence that converges, then it would contradict 1), which states that every convergent sequence is a Cauchy sequence.