I only have some confusion with the following:
I)We say (X,d) is a complete metric space iff every Cauchy sequence in X converges to some x in X.
I know that every Cauchy sequence in R (real numbers) is convergent and by the above statement R is complete metric space. Now if we consider the open interval A=(-1,1), which is a subset of R, and the sequence a_n=(1+1/n), n>0 it converges to 1 and 1 not in A, and that would mean by I A is not complete metric space. So does that mean every subset of a complete metric space does not have to be complete metric space?
Second Could anyone please provide me another example of a Cauchy sequence in another metric spaces which is not convergent?