Fixed radius disjoint ball covering vs. open covering
My professor posted the following problem, and I'm at a loss for where to begin. This is an introduction Real Analysis course, so I'm expecting the answer to be blatantly obvious. In fact, for small cases, I can see that it is true, but I am not sure how to prove it for the general case.
Notation: M_r(p) means the open neighborhood of radius r about p in the metric space M (this is the notation used by the professor).
X7. Suppose M is a sequentially compact metric space and let be a positive number. Then any sequence of disjoint balls in M of radius must be finite.
Prove this by contradiction: Suppose there is an infinite sequence M(x_n) of such balls. Can this sequence x_n of the centers of these balls have a convergent subsequence?
I solved this question with little difficulty.
X8. Problem X7 was used to prove the equivalence of sequential compactness and covering compactness for metric spaces. Using covering compactness we can strengthen X7 as follows:
Suppose M is a compact metric space and let be a positive number. Then there is an integer N, depending only on , so that any sequence of disjoint balls in M of radius has at most N elements.
Prove this as follows:
(a) Start with the collection of all open balls in M of radius , and show that there is a finite collection of balls of radius which covers M. Choose such a collection: M_ (c_n) for 1 <= n <= N.
(b) Now suppose M_ (x_k), for 1 <= k <= K, is any collection of disjoint balls of radius . Show that, for each k, there is an index n(k) so that c_n(k) is an element of M_ (x_k). Why is n(k) not equal to n(j) if k not equal to j?
(c) Why is K less than or equal to N?
For part (a), I'm having trouble showing that if you know that any sequence of disjoint balls of fixed radius in M can have a maximum of N elements, then there exists a finite collection of N open balls of that fixed radius in M such that the collection is an open cover for M. At least, I think that is what the question is asking, and intuitively, it seems correct. Starting with the finite cases, it seems quite plausible, making me wonder where I can apply the triangle inequality and hoping that some induction can prove this easily. Unfortunately, such a proof seems elusive at the moment, so any guidance on where to begin would be helpful.