p-complete spectra are p-local

Proof. Since \(X\) is \(p\)-complete, the localization map \(X \to X^{\wedge }_p\) is an \(\S /p\)-equivalence between \(\S /p\)-local spectra, hence is an isomorphism by Lemma 7.1.6. By Corollary 7.3.6, we may therefore write \(X \cong \lim _n(X/p^n)\). Each \(X/p^n\) is \(p\)-local by Proposition 7.2.12. By Lemma 7.2.8, \(p\)-local spectra are \(\S [P^{-1}]\)-local for \(P\) the set of all primes different from \(p\), so they are closed under limits by Exercise 7.1.4. □