Incomplete metric space .

Question

Proof that metric space of rational number with usual metric i.e $(\mathbb Q,d)$ is incomplete.

Attempt:- since we have a sequence of rational number $(1+1/n)^n$ converges to $e$, which is Cauchy but do not converge in $\mathbb Q$ (since $e$ is irrational ).

But how will I prove that this is Cauchy sequence! Can I use the fact that every convergent sequence is Cauchy? If yes, then the doubt is that the above sequence is convergent but not in $\mathbb Q$ and since when we say a Cauchy is not convergent we mean that it's is not convergent in the particular set ! So can we really use this fact of convergent implies Cauchy here !

Answer 1

The fact that a sequence is a Cauchy sequence only depends upon the distances between its elements. So, if it is a Cauchy sequence in $\mathbb R$, it is also a Cauchy sequence in $\mathbb Q$. And it is a Cauchy sequence in $\mathbb R$ since it converges in $\mathbb R$.

Answer 2

Let's say you have a sequence of rational numbers $a_n$ that is convergent in $\mathbb R$. Then it is a Cauchy sequence, meaning for any $\epsilon$ there exists $N$ such that $|a_m-a_n|<\epsilon$ for all $m, n>N$. These inequalities still hold in $\mathbb Q$, so while it may not converge in $\mathbb Q$, it is still Cauchy.