Let $(X,d)$ be a metric space. If $K⊆X$, and $K$
is a closed set. Does that mean any Cauchy sequence in $K$ converges in $K$?
If no, could someone give an example?
If yes, then what is the difference between complete sets and closed sets?
[Math] Closed Sets VS. Complete Sets
cauchy-sequencesreal-analysis
Best Answer
The rationals are closed in the rationals, but obviously not complete. For strict inclusion, take $\mathbb{Q} \cap [0,1]$ in $\mathbb{Q}$. Also not complete, but closed.
Incidentally, complete implies closed. Just look at limit points and sequences.