As an addendum, I am not looking for anyone to solve this problem for me. I am just looking for some guidance in the direction in which I should focus my thoughts. For instance, if looking into a potential Lebesgue number would help, I would be quite appreciative. On the other hand, if the problem can easily be solved with judicious use of the triangle inequality, or perhaps something equally ubiquitous, I would appreciate some advice for which distances to compare. For instance, I considered attempting to prove it by contradiction, suggesting that every collection of N balls of radius possesses points which are not covered, then I wanted to show for the finite case, where N=1, that this was a contradiction, then use induction to show that it is always a contradiction. So, I began with this rough sketch of a proof:
Because any sequence of disjoint balls of radius in M can have at most one element, the maximum distance between any two points in M must be less than 2 . Otherwise, let x,y be points in M such that d_M(x,y)>=2 . Because a ball of radius about x and y both would be disjoint, implying that there exists a sequence of two disjoint balls of radius . I get stuck here, as I am not sure if I need additional cases to show that the base case is true, or if I have enough here to demonstrate that it is. Either there exists points in between the furthest two points, or there are none. If there are points between, then centering the ball on them would suffice for an open cover. Alternatively, if the points x,y are disjoint, they must be within distance from each other, or within a distance of from a single point p in M, otherwise, again, an additional ball could be added to the sequence, but we know that 1 is the maximum number of disjoint balls allowed.
Then, for the induction step, I'm completely at a loss for how to proceed.