real number system 10 (Cauchy sequence)

In nested intervals, let $a_n$  be the increasing sequence from the left-hand side and $b_n$  be the decreasing sequence from the right-hand side. In the definition of the convergence the crucial condition was $|a_n-b_n|\rightarrow 0$  as $n\rightarrow \infty$ .

Given a sequence $c_n$  and consider a sequence $n_i$  of positive integers such that  $n_1<n_2<\cdots$ . Then the sequence $c_{n_i}$  is called a sub sequence of $c_n$ .
It is clear that $c_n$  converges to $c$  if and only if every sub sequence of $c_n$  converges to $c$ .

A sequence $c_n$  is said to be a Cauchy sequence if for any $\epsilon>0$ there is a $N$  such that $|c_n-c_m|<\epsilon$  if $n,m\geq N$ . (It means $|c_n-c_m|\rightarrow 0$ but $[c_n,c_m]$ are not always nested. )

We need to know the theorem in which every convergent sequence is a Cauchy sequence. However in the theorem the limit is not explicitly involved.