Is X Separable in a Totally Bounded Metric Space?

[tylerc1991]

Homework Statement

Let (X,d) be a totally bounded metric space. Prove that X is separable

Note: the definition of an epsilon-net in my book is this: A finite subset F of X is called an epsilon-net if for each x in X there is a p in F such that d(x,p) < epsilon

The Attempt at a Solution

X is separable if it has a countable dense subset.

(X,d) is totally bounded => for epsilon > 0, there is an epsilon-net for X
let D_n be the 1/n net for X
let A = (union) D_n ; n is an element of N (natural numbers)

A is obviously countable

The question is this, does A = X? if so, then I am done since the closure of A = X and hence A is dense. I would greatly appreciate it if someone could help me decide if A = X. Thank you!

 

[mighty2000]

I can confirm that your proof is correct. You have shown that A is a countable dense subset of X, which means that X is separable. Your method of constructing A using the epsilon-nets is a commonly used approach in mathematics. Good job!