Purpose of separating cauchy and convergent sequences

Question

I just discovered the definition of a cauchy sequence, and from what I've understood yet: - every cauchy sequence is convergent - every convergent sequence is cauchy

So I fail to understand what is the utility to call them any differently than simply convergent? Couldn't we simply use the cauchy sequences' definition as a second convergent sequences' definition or what am I missing? Because I know there are for instance 2 distinct definitions for the floor value of x, but it doesn't make us to use 2 different names.

Thank you~

Answer 1

It is not true that every Cauchy sequence is convergent. That is only true in a special class of spaces, called "complete" spaces.

It happens that the space of real numbers is complete, and so every sequence of real numbers converges if and only if it is Cauchy. But, this is not true in general.

For instance: consider the space $\mathbb{Q}$. Let $x_n$ be $\sqrt{2}$, truncated after $n$ decimal places. Then $(x_n)$ is a sequence in $\mathbb{Q}$, which is easily seen to be Cauchy, but which does not have a limit in $\mathbb{Q}$.

Answer 2

The convergence of Cauchy sequence give us an alternative way for the construction of $\mathbb{R}$. But, this equivalence is true in complete spaces. $\mathbb{R}$ is a complete metric space.