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A simple way to answer this question is: denote by $Spec$ the homotopy category of spectra, $Spec _0$ the category of (-1)-connected spectra, $InfL$ be the homotopy category of infinite loop spaces. The assertion that "connective spectra are the same as infinite loop space" simply means that the composition of functors $$Spec \to Spec _0 \stackrel{\Omega ^{\infty}}{\to} InfL$$ is an equivalence.