We prove that if an extensionR⊆Tof commutative rings satisfies the going-up property (for instance, ifTis an integral extension ofR), then any increasing chain of prime ideals ofR(indexed by an arbitrary linearly ordered set) is covered by some corresponding chain of prime ideals ofT. As a corollary, we recover the recent result of Kang and Oh that any such chain of prime ideals of an integral domainDis covered by a corresponding chain in some valuation overring ofD.