The sequence $\{a_n\}_{n=1}^\infty$ is cauchy if for every $\epsilon>0$, there is a corresponding natural number $N$ such that
$$
m,n\geq N\Rightarrow |a_m-a_n|<\epsilon
$$
I am doing a particular problem where the problem talks about cauchy sequences of rational numbers and I am not sure how that is different from a normal cauchy sequence (defined above).
If $\{a_n\}_{n=1}^\infty$ is a cauchy sequence in rational number and if there is a sub-sequence of this sequence, $\{a_{n_j}\}_{j=1}^\infty$ which converges to a rational number $\frac{p}{q}$, then I need to show that the sequence $\{a_n\}_{n=1}^\infty$ converges to the rational number $\frac{p}{q}$.
How this would be different if we did not talk about rational numbers so the problem was the following:
If $\{a_n\}_{n=1}^\infty$ is a cauchy sequence of real numbers and if there is a sub-sequence of this sequence, $\{a_{n_j}\}_{j=1}^\infty$ which converges to a real number $L$, then I need to show that the sequence $\{a_n\}_{n=1}^\infty$ converges to the real number $L$.
Since this question talks about rational numbers and not real numbers, it confuses me.
Best Answer
The main difference is that $\Bbb R$ is complete, while $\Bbb Q$ is not, which means that every Cauchy sequence of real numbers is convergent, while Cauchy sequence of rationals does not need to converge in $\Bbb Q$.
Thus, when you have your problem stated for real Cauchy sequences, it is almost trivial: sequence $(a_n)$ is Cauchy, therefore convergent, and since its subsequence has limit $L$, $(a_n)$ converges to $L$ as well.
On the other hand, if you switch back to rationals, you do not know that $(a_n)$ is convergent a priori. However, here comes the most important part: $\Bbb R$ is metric space completion of $\Bbb Q$ - meaning that every Cauchy sequence in $\Bbb Q$ will converge to some real number and any real number is a limit of Cauchy sequence in $\Bbb Q$.
Of course, the whole story about $\Bbb Q$ and $\Bbb R$ can be completely bypassed (but you specifically asked for the difference) and one can show that Cauchy sequence is convergent if and only if it has a convergent subsequence.