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**platinumpimp68plus1** You can show that the subcovers are finite by finding an arbitrary one.

You can argue:

At the point 0 for some large N, we have that 1/N<L, L some element of K. ie. 0 is a limit point of K. Pick a subcover that covers 0 and 1/N. Note that this covers infinitely many points between 0 and 1/N. That means that you only have finitely many points remaining: ie. 1, 1/2, 1/3, 1/4, 1/5,....1/(N+1). Pick one subcover to cover each of these points. Now you have a finite set of subcovers.