Proving Heine-Borel Theorem: A Compactness Characterization in Metric Spaces

[quasar987]

Homework Statement

In my book, they first prove Bolzano-Weierstrass (compact iff sequentially compact), then they prove a characterization of compactness for metric spaces: "A metric space M is compact iff it is complete and totally bounded." As a corollary of this, we get yet another characterization of compactness: "A subset A of a complete metric M space is compact iff it is closed in M and totally bounded."

Then they embark on the adventure of proving Heine-Borel (a subset of R^n is compact iff it is closed and bounded). They go about this in two ways. The first way sets out to prove the result using only the topological definition of compactness. After 3 lemmas the job is done. The second way uses the characterization by sequences (B-W) and after a dozen lines of reasoning, the job is done.

But it seems to me that they have left out the easiest and perhaps more enlightening way. Indeed, we already know that the metric space R^n is complete. Therefor, by the corollary I talked about, it suffices to show that in R^n, bounded ==> totally bounded. This way is more enlightening IMO because it actually shows that H-B is not a result that is restricted to R^n as the statement of the thm leads to believe. Rather, it suffices that a metric space M be complete that the the metric be not so exotic as to make a distinction btw bounded and totally bounded!

Or am I mistaken somewhere?

 

[morphism]

You're not mistaken. In fact, the corollary you've given is sometimes called the "metric space Heine-Borel theorem."