78 ¶ 399—must still be classified as a “private” mobile service, broadband users may no longer experience “the same Internet openness protections no matter what technology they use to access the Internet.” Id. at 5638 ¶ 92. C. Mobile petitioners also challenge the sufficiency of the Commission’s notice, particularly with respect to its redefinition of the public switched network as well as its removal of the word “all” from the definition of interconnected service. As noted above, the APA requires that an NPRM “include . . . either the terms or substance of the proposed rule or a description of the subjects and issues involved.” 5 U.S.C. § 553(b). But the APA also requires us to take “due account” of “the rule of prejudicial error.” Id. § 706. A deficiency of notice is harmless if the challengers had actual notice of the final rule, Small Refiner Lead PhaseDown Task Force v. EPA, 705 F.2d 506, 549 (D.C. Cir. 1983), or if they cannot show prejudice in the form of arguments they would have presented to the agency if given a chance, Owner-Operator Independent Drivers Ass’n v. Federal Motor Carrier Safety Administration, 494 F.3d 188, 202 (D.C. Cir. 2007). Both circumstances are present here, and each independently supports our conclusion that any lack of notice was ultimately harmless. As such, we need not decide whether the Commission gave adequate notice of its redefinition of the public switched network in the NPRM. As mobile petitioners acknowledge, Vonage raised the idea of redefining the public switched network in its comments, pointing out the Commission’s “authority to interpret the key terms in th[e] definition [of commercial mobile service], including ‘interconnected’ and ‘public