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 ?