1. ## Bolzano Weierstrass proof

I have been asked to prove that a metric space has the Bolzano Weierstrass property if and only if it is complete and totally bounded.

It seems obvious to me that Bolzano Weierstrass implies completeness however unless I am mistaken the real line is a counterexample (with the Euclidean metric) to the statement.

R is not totally bounded and it satisfies the Bolzano Weierstrass property. Am I being stupid here? It is just that I was trying to do a past-paper question and this is what I am being asked to show.

2. Originally Posted by ihateyouall
I have been asked to prove that a metric space has the Bolzano Weierstrass property if and only if it is complete and totally bounded.

It seems obvious to me that Bolzano Weierstrass implies completeness however unless I am mistaken the real line is a counterexample (with the Euclidean metric) to the statement.

R is not totally bounded and it satisfies the Bolzano Weierstrass property. Am I being stupid here? It is just that I was trying to do a past-paper question and this is what I am being asked to show.
The Bolzano–Weierstrass property, otherwise known as sequential compactness, is the property that every sequence has a convergent subsequence. The real line does not have this property. For example, the sequence of natural numbers does not have a convergent subsequence.

3. Thanks. I knew that I must be wrong about something.