Totally Bounded & Completeness

[delta_epsilon]

Homework Statement

I haven't been able to find any theorems stating the relationship between a totally bounded space and a complete metric space, i.e., whether totally boundedness implies completeness. (I know that completeness implies totally boundedness though). Is it true that totally boundedness implies completeness?

Any help is much appreciated.

Homework Equations

A metric space is totally bounded if give any ε>0, there exist finitely many points $x_1, ... ,x_n$ $\in$ M s.t. A $\subset$ $\bigcup_{i=1}^{n}$ $B_{\epsilon} (x_i)$ where n is finite.

A metric space M is said to be complete if every Cauchy sequence in M converges to a point in M.

 

[lippyka]

The Attempt at a SolutionNo, totally boundedness does not imply completeness. A counterexample is the set of rational numbers with respect to the Euclidean metric. This set is totally bounded, but it is not complete since the sequence of rational numbers approaching a given irrational number does not converge to a point in the set.