Show that in any metric space, a compact set is bounded. Solution In any metric space, a compact set is bounded. Proof: Let a be any point in a compact set C. If we consider the open balls around a, i.e. centred at a and with any radius. This can cover any set as it is with any radius, because all points in the set are a distance away from a, by the definition of metric space. Any finite subcover of this cover must be bounded. The reason is all balls in the subcover are contained in the largest open ball within that subcover. Therefore, any set covered by this subcover must also be bounded. In other words, in any metric space, a compact set is bounded..