Complete subspace of connected space

Question

Let $A$ be a connected metric space and $B$ subspace such that $B$ and $A\backslash B$ are complete.

How do I show that $B$ is either empty or equal to $A$?

What I thought:
We need to show that $B$ not empty implies $B=A$.
$B$ is complete, so each Cauchy sequence converges in $B$. Further $A$ has exactly two clopens, namely $A$ and empty.
I think that we need to show that $B$ is clopen. How do I do that?

Answer 1

If $B$ is complete then $B$ has to be closed. But that is true for $A\setminus B$ too. So $B$ and $A\setminus B$ are clopen.