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 !