p-completion preserves cofiber sequences

Question

Suppose I have a cofiber sequence $X \to Y \to Z$ of spectra in the stable homotopy category. I want to show that there is still a cofiber sequence $X^\wedge_p \to Y^\wedge_p \to Z^\wedge_p$ after p-completing, but I am a little lost. I tried looking at Bousfield's paper "The localization of spectra with resepct to homology", where he proves that a model for the p-completion is given by the localization at the Moore spectrum $S\mathbb{Z}/p$. This localization is equivalent to the function spectrum $F(\Sigma^{-1}S\mathbb{Z}/p^\infty, X)$. In showing this, Bousfield provides a nice cofiber sequence $$F(S\mathbb{Z}[p^{-1}], X) \to F(S\mathbb{Z}, X) \to X^\wedge_p.$$ My idea for a proof involved interlacing the cofiber sequence above for $X, Y, Z$, and the octahedral axiom will give the result. My problem is that this hinges on the functor $F(\Sigma^{-1}S\mathbb{Z}/p^{\infty}, -)$ taking cofiber sequences to cofiber sequences. I do not think this is true. Does anyone have a hint in the right direction? Maybe I should leverage defining the p-completion as a limit instead.