I have been reading the metric-space part of John B. Conway's A Course in Point-set Topology, and the definition of convergence in metric spaces seems ambiguous to me.
The author does not make it clear whether the point $x = \lim_n x_n$ in this definition should fall in $X$ or not. And it seems to me that given some context $x$ could be out of the given metric space, like in this one (Proposition 1.2.7), $x$ need not be in $A$:
But in other part of the book, I find it better to admit that the limit is in the given metric space, e.g. in the following definition of complete metric space:
That is because, if in Definition 1.2.10 the limit of Cauchy sequence $\{x_n\}$ is allowed to be outside the space $X$, then I feel (but not quite sure) the following proposition does not make sense, since I guess to establish the necessity one has to use the hypothesis that the limit of any Cauchy sequence falls in $Y$:
The question may sound naive, but I really get confused about the seeming inconsistence of the text. Can anyone please explain it to me? Thanks in advance!
edit: If there is only one space in the context, things are clear. But in the two propositions I refer to, there is a underlying space X and its subspaces, respectively denoted as A and Y, and the sequences in both cases are in the subspace. It seems in the former case, the text means 'the limit need not be in the subspace A'; but in the latter the text means 'the limit must lie in the subspace Y'. That's what confused me.