Difference in the definitions of cauchy sequence in Real Sequence and in Metric space.

Question

I have read two definitions of Cauchy sequence with a result.

In real sequence:

A sequence {$a_n$} is called a Cauchy sequence if for each $\epsilon >0$, there exists a positive integer m such that $|a_{n_1} - a_{n_2}| < \epsilon$, $\forall n_1,n_2 \geq m$

Result: A sequence is convergent iff it is Cauchy sequence.

In metric space:

A sequence {$a_n$} of points of $(X,d)$ is called a Cauchy sequence if for each $\epsilon >0$, there exists a positive integer $n_0$ such that $d(x_n,x_m) < \epsilon$, $\forall n,m \geq n_0$

Result: Every convergent sequence is a Cauchy sequence but converse is not true. Counter example: Consider a space $X = (0,1]$ with usual metric space. Take {$a_n$} = {$\frac{1}{n}$} is a Cauchy sequence converges to 0 but $0 \notin X$

My question is that: There is a difference between these results, why ?

Answer 1

On some metric spaces, every Cauchy sequence converges (and on every metric space, every convergent sequence is a Cauchy sequence). And $\Bbb R$, endowed with its usual distance, is one of the spaces. By the way, these are the complete metric spaces. And $(0,1]$ (again, endowed with its usual distance) is not complete.