# Thread: The discrete Metric- Completeness and Compactness

1. ## The discrete Metric- Completeness and Compactness

Hello, I'm just working through this problem but I am a little shaky with some of the definitions

Let M be a set and let d be the discrete metric on
M.
1. Show that (M, d) is complete.
2. Describe all the compact subsets of M.

Since (M,d) is equipped with the discrete metric all sequences inside that metric converge to some pt that repeats. How is this cauchy?

2. Originally Posted by Scopur
Hello, I'm just working through this problem but I am a little shaky with some of the definitions

Let M be a set and let d be the discrete metric on
M.
1. Show that (M, d) is complete.
2. Describe all the compact subsets of M.

Since (M,d) is equipped with the discrete metric all sequences inside that metric converge to some pt that repeats. How is this cauchy?
hm... every converging sequence is Cauchy sequence

Th : "every compact metric space is complete .... "

if X is compact metric space and let $(x_n)$ be Cauchy sequence in X. because of compactness there is sub sequence $x_{nk}$ of our sequence.. and sub sequence is converging $x_{nk} \to x_0 \in X \;\; (k \to \infty )$ now we have

$d(x_n,x_0) <= d(x_n, x_{nk} ) + d ( x_{nk} , x_0)$

now first on the right side can be made very small because it is Cauchy , and second to because of convergence of the sub sequence so we have

$d(x_n,x_0) \to 0 \;\; (n\to \infty)$

meaning sequence $x_n$ is converging and space X is complete

3. for part b) use the fact that given any open cover of a compact set, there exists a finite subcover. Open balls of radius one about x, are denoted $\ B_1 (x)$ and $\forall x \in M$ , $\ B_1 (x) = \{x\}$
so only finite subsets are compact.