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.
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, $\displaystyle M$ may not be countable, so this doesn't work. Try proving that if $\displaystyle (x_n)$ is a Cauchy sequence then $\displaystyle X=\overline{\{x_n:n\in\mathbb{N}\}}$ contains at most one more element and thus is also countable. But, then $\displaystyle (x_n)$ is a Cauchy sequence in the countable closed subset $\displaystyle X\subseteq M$.