Sequential characterization of closedness, and completeness of a set.

Question

After reading the theroem about Sequential characterization of closedness of the set, and the definition of a complete set(a metric space is said to be complete if every Cauchy sequence has its limit in the space X ), I can't undertand what's the difference between both claimings.

Answer 1

My guess is that the claims are these:

• $S$ is closed if whenever a sequence $(x_n)_{n\in\mathbb N}$ of elements of $S$ converges, $\lim_nx_n\in S$;
• $S$ is complete if every Cauchy sequence of elements of $S$ converges to an element of $S$.

The difference is that in the case of closed sets we are assuming that the sequence is convergent, whereas in the case of complete sets we are assuming that the sequence is a Cauchy sequence. These are distinct assumptions.