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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 ?

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  • $\begingroup$ Regarding your title, notice that if you compare your two definitions of Cauchy sequence in $\mathbb R$ versus in an arbitrary metric space, by using the distance formula $d(x_n,x_m) = |x_n-x_m|$ in $\mathbb R$ you will see that there is utterly no difference between those two definitions that you wrote (except for some cosmetic changes in bound variables). So, as you say in your very last sentence, the difference lies in the results or theorems about Cauchy sequences in $\mathbb R$ versus a general metric space. $\endgroup$
    – Lee Mosher
    Jun 24 at 23:08
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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.

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