1. prove M is complete

Suppose that every countable, closed subset of M is complete. Prove that M is complete.

my idea is to show that M is closed first, then let (Xn) be a cauchy sequence in M, so it's cauchy in all of its subset and then converge, so M is complete.

2. Re: prove M is complete

Originally Posted by wopashui
Suppose that every countable, closed subset of M is complete. Prove that M is complete.

my idea is to show that M is closed first, then let (Xn) be a cauchy sequence in M, so it's cauchy in all of its subset and then converge, so M is complete.
But, $M$ may not be countable, so this doesn't work. Try proving that if $(x_n)$ is a Cauchy sequence then $X=\overline{\{x_n:n\in\mathbb{N}\}}$ contains at most one more element and thus is also countable. But, then $(x_n)$ is a Cauchy sequence in the countable closed subset $X\subseteq M$.

3. Re: prove M is complete

if A is subset of M, and (Xn) is cauchy in M, does it imply (Xn) is cauchy in A?

4. Re: prove M is complete

Originally Posted by wopashui
if A is subset of M, and (Xn) is cauchy in M, does it imply (Xn) is cauchy in A?
Yes. But, the problem is that you need to prove you can always put $(x_n)$ inside some COUNTABLE closed subset.