 The space "M" is called precompact or totally bounded if for every there exist finitely many open balls of radius "r" whose union covers "M".Since the set of the centres of these balls is finite, it has finite diameter, from which it follows (using the triangle inequality) that every totally bounded space is bounded.The converse does not hold, since any infinite set can be given the discrete metric (one of the examples above) under which it is bounded and yet not totally bounded.