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We verify a conjecture of Rognes by establishing a localization cofiber sequence of spectra for the algebraic K-theory of topological K-theory. We deduce the existence of this sequence as a consequence of a dévissage theorem identifying the K-theory of the Waldhausen category of finitely generated finite stage Postnikov towers of modules over a connective ring spectrum R with the Quillen K-theory of the abelian category of finitely generated -modules.