Let $(X,d)$ - metric space such that $A,B \subset X$ and $ X=A\cup B$ where $(A,d_{|A\times A}), (B,d_{|B\times B})$ are complete subspaces. Prove that $(X,d)$ is complete.
I think it should be obvious that the sum of the complete subspaces is complete, but nevertheless a few years ago this task was in an exam so I would like to know how to prove it.
Let $(x_n)_n$ be a Cauchy sequence in $X$. We want to show that $x_n$ converges. As $X=A\cup B$ infinitely many elements of $(x_n)_n$ and hence a subsequence have to be in one of the sets $A,B$. Without loss of generality assume that there is a subsequence $(x_{n_k})_k$ of $(x_n)_n$ contained in $A$. As subsequences of Cauchy sequences are again Cauchy, we see that $(x_{n_k})_k$ is a Cauchy sequence. By assumption $A$ is complete, hence $(x_{n_k})_k$ is convergent. This then implies that $(x_n)$ is convergent (In general a Cauchy sequence with a convergent subsequence is itself convergent). Hence $X$ is complete.